Recent zbMATH articles in MSC 60https://zbmath.org/atom/cc/602021-11-25T18:46:10.358925ZWerkzeugBooks review of: P. Müller et al., Bayesian nonparametric data analysishttps://zbmath.org/1472.000092021-11-25T18:46:10.358925Z"Bouza, C. N."https://zbmath.org/authors/?q=ai:bouza-herrera.carlos-narcisoReview of [Zbl 1333.62003].Book review of: Á. Cartea et al., Algorithmic and high-frequency tradinghttps://zbmath.org/1472.000312021-11-25T18:46:10.358925Z"Rosenbaum, Mathieu"https://zbmath.org/authors/?q=ai:rosenbaum.mathieuReview of [Zbl 1332.91001].Statistical structure of concave compositionshttps://zbmath.org/1472.050162021-11-25T18:46:10.358925Z"Dalal, Avinash J."https://zbmath.org/authors/?q=ai:dalal.avinash-j"Lohss, Amanda"https://zbmath.org/authors/?q=ai:lohss.amanda"Parry, Daniel"https://zbmath.org/authors/?q=ai:parry.daniel-tSummary: In this paper, we study concave compositions, an extension of partitions that were considered by \textit{G. E. Andrews} et al. [Algebra Number Theory 7, No. 9, 2103--2139 (2013; Zbl 1282.05016)]. They presented several open problems regarding the statistical structure of concave compositions including the distribution of the perimeter and tilt, the number of summands, and the shape of the graph of a typical concave composition. We present solutions to these problems by applying Fristedt's conditioning device on the uniform measure.Sampling hypergraphs with given degreeshttps://zbmath.org/1472.051172021-11-25T18:46:10.358925Z"Dyer, Martin"https://zbmath.org/authors/?q=ai:dyer.martin-e"Greenhill, Catherine"https://zbmath.org/authors/?q=ai:greenhill.catherine-s"Kleer, Pieter"https://zbmath.org/authors/?q=ai:kleer.pieter"Ross, James"https://zbmath.org/authors/?q=ai:ross.james-lance|ross.james-e|ross.james-b"Stougie, Leen"https://zbmath.org/authors/?q=ai:stougie.leenSummary: There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm \(\mathcal{A}\) for sampling bipartite graphs with given degrees, uniformly or nearly uniformly, in (expected) polynomial time. The expected runtime of the hypergraph sampling algorithm depends on the (expected) runtime of the bipartite graph sampling algorithm \(\mathcal{A} \), and the probability that a uniformly random bipartite graph with given degrees corresponds to a simple hypergraph. We give some conditions on the hypergraph degree sequence which guarantee that this probability is bounded below by a positive constant.A general stochastic matching model on multigraphshttps://zbmath.org/1472.051232021-11-25T18:46:10.358925Z"Begeot, Jocelyn"https://zbmath.org/authors/?q=ai:begeot.jocelyn"Marcovici, Irène"https://zbmath.org/authors/?q=ai:marcovici.irene"Moyal, Pascal"https://zbmath.org/authors/?q=ai:moyal.pascal"Rahme, Youssef"https://zbmath.org/authors/?q=ai:rahme.youssefSummary: We extend the general stochastic matching model on graphs introduced in \textit{J. Mairesse} and \textit{P. Moyal} [J. Appl. Probab. 53, No. 4, 1064--1077 (2016; Zbl 1356.60147)], to matching models on multigraphs, that is, graphs with self-loops. The evolution of the model can be described by a discrete time Markov chain whose positive recurrence is investigated. Necessary and sufficient stability conditions are provided, together with the explicit form of the stationary probability in the case where the matching policy is `First Come, First Matched'.Thresholds versus fractional expectation-thresholdshttps://zbmath.org/1472.051322021-11-25T18:46:10.358925Z"Frankston, Keith"https://zbmath.org/authors/?q=ai:frankston.keith"Kahn, Jeff"https://zbmath.org/authors/?q=ai:kahn.jeff-d"Narayanan, Bhargav"https://zbmath.org/authors/?q=ai:narayanan.bhargav-p"Park, Jinyoung"https://zbmath.org/authors/?q=ai:park.jinyoungSummary: Proving a conjecture of \textit{M. Talagrand} [in: Proceedings of the 42nd annual ACM symposium on theory of computing, STOC '10. Cambridge, MA, USA, June 5--8, 2010. New York, NY: Association for Computing Machinery (ACM). 13--36 (2010; Zbl 1293.60014)], a fractional version of the ``expectation-threshold'' conjecture of \textit{J. Kahn} and \textit{G. Kalai} [Comb. Probab. Comput. 16, No. 3, 495--502 (2007; Zbl 1118.05093)], we show that \(p_c(\mathcal{F})=O(q_f(\mathcal{F})\log\ell(\mathcal{F}))\) for any increasing family \(\mathcal{F}\) on a finite set \(X\), where \(p_c(\mathcal{F})\) and \(q_f(\mathcal{F})\) are the threshold and ``fractional expectation-threshold'' of \(\mathcal{F}\), and \(\ell(\mathcal{F})\) is the maximum size of a minimal member of \(\mathcal{F}\). This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings [\textit{A. Johansson} et al., Random Struct. Algorithms 33, No. 1, 1--28 (2008; Zbl 1146.05040)], bounded degree spanning trees [\textit{R. Montgomery}, Adv. Math. 356, Article ID 106793, 92 p. (2019; Zbl 1421.05080)], and bounded degree graphs (new). We also resolve (and vastly extend) the ``axial'' version of the random multi-dimensional assignment problem (earlier considered by Martin-Mézard-Rivoire and \textit{A. Frieze} and \textit{G. B. Sorkin} [Random Struct. Algorithms 46, No. 1, 160--196 (2015; Zbl 1347.60141)]). Our approach builds on a recent breakthrough of \textit{R. Alweiss} et al. [in: Proceedings of the 52nd annual ACM SIGACT symposium on theory of computing, STOC '20, Chicago, IL, USA, June 22--26, 2020. New York, NY: Association for Computing Machinery (ACM). 624--630 (2020; Zbl 07298275)] on the Erdő-Rado ``Sunflower Conjecture''.The degree analysis of an inhomogeneous growing network with two types of verticeshttps://zbmath.org/1472.051392021-11-25T18:46:10.358925Z"Huang, Huilin"https://zbmath.org/authors/?q=ai:huang.huilinSummary: We consider an inhomogeneous growing network with two types of vertices. The degree sequences of two different types of vertices are investigated, respectively. We not only prove that the asymptotical degree distribution of type \(s\) for this process is power law with exponent \(2 + \left(\left(1 + \delta\right) q_s + \beta \left(1 - q_s\right)\right) / \alpha q_s\), but also give the strong law of large numbers for degree sequences of two different types of vertices by using a different method instead of Azuma's inequality. Then we determine asymptotically the joint probability distribution of degree for pairs of adjacent vertices with the same type and with different types, respectively.Networks beyond pairwise interactions: structure and dynamicshttps://zbmath.org/1472.051432021-11-25T18:46:10.358925Z"Battiston, Federico"https://zbmath.org/authors/?q=ai:battiston.federico"Cencetti, Giulia"https://zbmath.org/authors/?q=ai:cencetti.giulia"Iacopini, Iacopo"https://zbmath.org/authors/?q=ai:iacopini.iacopo"Latora, Vito"https://zbmath.org/authors/?q=ai:latora.vito"Lucas, Maxime"https://zbmath.org/authors/?q=ai:lucas.maxime"Patania, Alice"https://zbmath.org/authors/?q=ai:patania.alice"Young, Jean-Gabriel"https://zbmath.org/authors/?q=ai:young.jean-gabriel"Petri, Giovanni"https://zbmath.org/authors/?q=ai:petri.giovanniSummary: The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a variety of complex systems has been successfully described as networks whose interacting pairs of nodes are connected by links. Yet, from human communications to chemical reactions and ecological systems, interactions can often occur in groups of three or more nodes and cannot be described simply in terms of dyads. Until recently little attention has been devoted to the higher-order architecture of real complex systems. However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can enhance our modeling capacities and help us understand and predict their dynamical behavior. Here we present a complete overview of the emerging field of networks beyond pairwise interactions. We discuss how to represent higher-order interactions and introduce the different frameworks used to describe higher-order systems, highlighting the links between the existing concepts and representations. We review the measures designed to characterize the structure of these systems and the models proposed to generate synthetic structures, such as random and growing bipartite graphs, hypergraphs and simplicial complexes. We introduce the rapidly growing research on higher-order dynamical systems and dynamical topology, discussing the relations between higher-order interactions and collective behavior. We focus in particular on new emergent phenomena characterizing dynamical processes, such as diffusion, synchronization, spreading, social dynamics and games, when extended beyond pairwise interactions. We conclude with a summary of empirical applications, and an outlook on current modeling and conceptual frontiers.Combinatorial anti-concentration inequalities, with applicationshttps://zbmath.org/1472.051472021-11-25T18:46:10.358925Z"Fox, Jacob"https://zbmath.org/authors/?q=ai:fox.jacob"Kwan, Matthew"https://zbmath.org/authors/?q=ai:kwan.matthew"Sauermann, Lisa"https://zbmath.org/authors/?q=ai:sauermann.lisaSummary: We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some ``Poisson-type'' anti-concentration theorems that give bounds of the form \(1/e+o(1)\) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős-Littlewood-Offord theorem and improves a theorem of \textit{R. Meka} et al. [Theory Comput. 12, Paper No. 11, 17 p. (2016; Zbl 1392.68193)] for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs.Clone-induced approximation algebras of Bernoulli distributionshttps://zbmath.org/1472.080052021-11-25T18:46:10.358925Z"Yashunsky, Alexey D."https://zbmath.org/authors/?q=ai:yashunsky.aleksey-dSummary: We consider the problem of approximating distributions of Bernoulli random variables by applying Boolean functions to independent random variables with distributions from a given set. For a set \(B\) of Boolean functions, the set of approximable distributions forms an algebra, named the approximation algebra of Bernoulli distributions induced by \(B\). We provide a complete description of approximation algebras induced by most clones of Boolean functions. For remaining clones, we prove a criterion for approximation algebras and a property of algebras that are finitely generated.The distribution of the maximum of partial sums of Kloosterman sums and other trace functionshttps://zbmath.org/1472.112172021-11-25T18:46:10.358925Z"Autissier, Pascal"https://zbmath.org/authors/?q=ai:autissier.pascal"Bonolis, Dante"https://zbmath.org/authors/?q=ai:bonolis.dante"Lamzouri, Youness"https://zbmath.org/authors/?q=ai:lamzouri.younessLet \(\mathcal{F}=\{\varphi_a\}_{a\in\Omega_m}\) be a family of periodic functions, where \(\Omega_m\) is a non-empty finite set, and for each \(a\in\Omega_m\), \(\varphi_a:\mathbb{Z}\to\mathbb{C}\) is \(m\)-periodic and its Fourier transform \(\widehat{\varphi_a}\) is real-valued and uniformly bounded. For a positive real number \(V\), distribution of the maximum of partial sums of families of \(m\)-periodic complex-valued functions is defined by
\[
\Phi_{\mathcal{F}}(V)=\frac{1}{\#\Omega_m}\,\#\left\{a\in\Omega_m : \frac{1}{\sqrt{m}}\max_{x<m}\left|\sum_{0\leq n\leq x}\varphi_a(n)\right|>V\right\}.
\]
In the paper under review, assuming certain conditions, the authors prove that there exists a constant \(B\) such that for all real numbers \(B\leq V\leq (N/\pi)(\log \log m - 2 \log \log \log m) - B\) one has
\[
\Phi_{\mathcal{F}}(V)=\exp\left(-\exp\left(\frac{\pi}{N}V+O(1)\right)\right).
\]
This general estimate covers some previously known results on the partial sums of character sums, Kloosterman sums and other families of \(\ell\)-adic trace functions.Moments of random multiplicative functions. I: Low moments, better than squareroot cancellation, and critical multiplicative chaoshttps://zbmath.org/1472.112542021-11-25T18:46:10.358925Z"Harper, Adam J."https://zbmath.org/authors/?q=ai:harper.adam-jSummary: We determine the order of magnitude of \(\mathbb{E}|\sum_{n\leqslant x}f(n)|^{2q}\), where \(f(n)\) is a Steinhaus or Rademacher random multiplicative function, and \(0\leqslant q\leqslant 1\). In the Steinhaus case, this is equivalent to determining the order of \(\lim_{T\rightarrow \infty }\frac{1}{T}\int_0^T|\sum_{n\leqslant x}n^{-it}|^{2q}\,dt\).
In particular, we find that \(\mathbb{E}|\sum_{n\leqslant x}f(n)|\asymp \sqrt{x}/(\log \log x)^{1/4} \). This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of \(\sum_{n\leqslant x}f(n)\).
The proofs develop a connection between \(\mathbb{E}|\sum_{n\leqslant x}f(n)|^{2q}\) and the \(q\) th moment of a critical, approximately Gaussian, multiplicative chaos and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.Singularity of random symmetric matrices -- simple proofhttps://zbmath.org/1472.150432021-11-25T18:46:10.358925Z"Ferber, Asaf"https://zbmath.org/authors/?q=ai:ferber.asafSummary: In this paper we give a simple, short, and self-contained proof for a non-trivial upper bound on the probability that a random \(\pm 1\) symmetric matrix is singular.On multiplicities of irreducibles in large tensor product of representations of simple Lie algebrashttps://zbmath.org/1472.170352021-11-25T18:46:10.358925Z"Postnova, Olga"https://zbmath.org/authors/?q=ai:postnova.olga-v"Reshetikhin, Nicolai"https://zbmath.org/authors/?q=ai:reshetikhin.nikolai-yuSummary: In this paper, we study the asymptotics of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of \textit{P. Biane} [C. R. Acad. Sci., Paris, Sér. I 316, No. 8, 849--852 (1993; Zbl 0805.17005)] and \textit{T. Tate} and \textit{S. Zelditch} [J. Funct. Anal. 217, No. 2, 402--447 (2004; Zbl 1062.22026)]. We also derive the asymptotics of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution.On a theorem of Avezhttps://zbmath.org/1472.200842021-11-25T18:46:10.358925Z"Elder, Murray"https://zbmath.org/authors/?q=ai:elder.murray-j"Rogers, Cameron"https://zbmath.org/authors/?q=ai:rogers.cameronSummary: For each symmetric, aperiodic probability measure \(\mu\) on a finitely generated group \(G\), we define a subset \(A_{\mu}\) consisting of group elements \(g\) for which the limit of the ratio \(\mu^{\ast n}(g)/\mu^{\ast n}(e)\) tends to 1. We prove that \(A_{\mu}\) is a subgroup, is amenable, contains every finite normal subgroup, and \(G=A_{\mu}\) if and only if \(G\) is amenable. For non-amenable groups we show that \(A_{\mu}\) is not always a normal subgroup and can depend on the measure. We formulate some conjectures relating \(A_{\mu}\) to the amenable radical.Geometric functionals of fractal percolationhttps://zbmath.org/1472.280082021-11-25T18:46:10.358925Z"Klatt, Michael A."https://zbmath.org/authors/?q=ai:klatt.michael-andreas"Winter, Steffen"https://zbmath.org/authors/?q=ai:winter.steffenSummary: Fractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system-spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the percolation thresholds have been approximated well using additive geometric functionals, known as intrinsic volumes. Motivated by the question of whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process \(F\). They arise as limits of expected functionals of finite approximations of \(F\). We establish the existence of these limit functionals and obtain explicit formulas for them as well as for their finite approximations.Positive harmonic functions on the Heisenberg group. IIhttps://zbmath.org/1472.310122021-11-25T18:46:10.358925Z"Benoist, Yves"https://zbmath.org/authors/?q=ai:benoist.yvesThis article discusses extremal positive harmonic functions for finitely supported measures on the discrete Heisenberg group. The author establishes that extremal positive harmonic functions are proportional either to characters or to translates of induced from characters.
For Part I, see [the author, ``Positive Harmonic Functions on the Heisenberg group. I'', Preprint, \url{arXiv:1907.05041}].A new generalized Gronwall inequality with a double singularity and its applications to fractional stochastic differential equationshttps://zbmath.org/1472.340112021-11-25T18:46:10.358925Z"Ding, Xiao-Li"https://zbmath.org/authors/?q=ai:ding.xiaoli"Daniel, Cao-Labora"https://zbmath.org/authors/?q=ai:daniel.cao-labora"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseThis paper focuses primarily on a new generalized Gronwall inequality with a double singularity and its applications to fractional stochastic differential equations
The first few pages introduce all the necessary notation and terminology to understand the paper: the theory of operators and ingenious techniques to investigate the well-posedness of mild solution to semilinear fractional stochastic differential equations is discussed.
The main results are presented in Section 3 (Generalized Gronwall inequalities with weakly singular kernels) with the help of Theorem 3.1 and 3.2. In Section 4, the properties and integral inequalities obtained in Section 3 to discuss the well-posedness of semilinear fractional stochastic differential equations are studied.Ergodicity and threshold behaviors of a predator-prey model in stochastic chemostat driven by regime switchinghttps://zbmath.org/1472.340982021-11-25T18:46:10.358925Z"Wang, Liang"https://zbmath.org/authors/?q=ai:wang.liang"Jiang, Daqing"https://zbmath.org/authors/?q=ai:jiang.daqingSummary: This paper deals with a stochastic predator-prey model in chemostat which is driven by Markov regime switching. For the asymptotic behaviors of this stochastic system, we establish the sufficient conditions for the existence of the stationary distribution. Then, we investigate, respectively, the extinction of the prey and predator populations. We explore the new critical numbers between survival and extinction for species of the dual-threshold chemostat model. Numerical simulations are accomplished to confirm our analytical conclusions.Corrigendum to: ``Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion''https://zbmath.org/1472.341122021-11-25T18:46:10.358925Z"Blouhi, T."https://zbmath.org/authors/?q=ai:blouhi.tayeb"Caraballo, T."https://zbmath.org/authors/?q=ai:caraballo.tomas"Ouahab, A."https://zbmath.org/authors/?q=ai:ouahab.abdelghaniSummary: In this paper we correct an error made in our paper [ibid. 34, No. 5, 792--834 (2016; Zbl 1380.34091)]. In fact, in this corrigendum we present the correct hypotheses and results, and highlight that the results can be proved using the same method used in the original work. The main feature is that we used a result which has been proved only when the diffusion term does not depend on the unknown.Stochastic homogenization of a class of quasiconvex viscous Hamilton-Jacobi equations in one space dimensionhttps://zbmath.org/1472.350332021-11-25T18:46:10.358925Z"Yilmaz, Atilla"https://zbmath.org/authors/?q=ai:yilmaz.atillaSummary: We prove homogenization for a class of viscous Hamilton-Jacobi equations in the stationary \& ergodic setting in one space dimension. Our assumptions include most notably the following: the Hamiltonian is of the form \(G(p) + \beta V(x, \omega)\), the function \(G\) is coercive and strictly quasiconvex, \( \min G = 0\), \(\beta > 0\), the random potential \(V\) takes values in \([0, 1]\) with full support and it satisfies a hill condition that involves the diffusion coefficient. Our approach is based on showing that, for every direction outside of a bounded interval \(( \theta_1(\beta), \theta_2(\beta))\), there is a unique sublinear corrector with certain properties. We obtain a formula for the effective Hamiltonian and deduce that it is coercive, identically equal to \(\beta\) on \(( \theta_1(\beta), \theta_2(\beta))\), and strictly monotone elsewhere.Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interactionhttps://zbmath.org/1472.350352021-11-25T18:46:10.358925Z"Wang, Xiaoli"https://zbmath.org/authors/?q=ai:wang.xiaoli.1"Shi, Junping"https://zbmath.org/authors/?q=ai:shi.junping"Zhang, Guohong"https://zbmath.org/authors/?q=ai:zhang.guohongA reaction-diffusion model describing water and plant interaction in a flat environment is studied. The system is governed by the Klausmeier-Gray-Scott equations
\begin{align*}
\frac{\partial W}{\partial t} &= d_1 \Delta W + a - W B^2 - W, \\
\frac{\partial B}{\partial t} &= d_2 \Delta B + W B^2 - mB
\end{align*}
with Neumann boundary conditions.
After carefully investigating the existence and stability of uniform steady states the authors study the bifurcations from these steady states to large amplitude spatial patterned solutions. In the case of small rain fall \(a\) the authors construct a simpler shadow system by considering the singular limit \(d_1\rightarrow \infty\). The authors also carry out numerical investigations on linear and rectangular domains, which demonstrate the occurence of complicated vegetation patterns.
The article shows, that slow plant diffusion and fast water diffusion can support a vegetation state with vegetation concentrating in a small area or ``spots'', even when the rainfall is too low to support uniform vegetation.Asymptotic behavior of density in the boundary-driven exclusion process on the Sierpinski gaskethttps://zbmath.org/1472.352082021-11-25T18:46:10.358925Z"Chen, Joe P."https://zbmath.org/authors/?q=ai:chen.joe-p-j|chen.joe-p"Gonçalves, Patrícia"https://zbmath.org/authors/?q=ai:goncalves.patricia-cSummary: We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.Spectral heat content for Lévy processeshttps://zbmath.org/1472.352092021-11-25T18:46:10.358925Z"Grzywny, Tomasz"https://zbmath.org/authors/?q=ai:grzywny.tomasz"Park, Hyunchul"https://zbmath.org/authors/?q=ai:park.hyunchul"Song, Renming"https://zbmath.org/authors/?q=ai:song.renmingSummary: In this paper we study the spectral heat content for various Lévy processes. We establish the small time asymptotic behavior of the spectral heat content for Lévy processes of bounded variation in \(\mathbb R^d, d \geq 1\). We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in \(\mathbb R\) with respect to symmetric Lévy processes of unbounded variation under certain conditions on their characteristic exponents. Finally, we establish that the small time asymptotic behavior of the spectral heat content is stable under integrable perturbations to the Lévy measure.Probabilistic representation for mild solution of the Navier-Stokes equationshttps://zbmath.org/1472.352722021-11-25T18:46:10.358925Z"Olivera, Christian"https://zbmath.org/authors/?q=ai:olivera.christianThe author extends the approach by \textit{P. Constantin} and \textit{G. Iyer} [Commun. Pure Appl. Math. 61, No. 3, 330--345 (2008; Zbl 1156.60048); Ann. Appl. Probab. 21, No. 4, 1466--1492 (2011; Zbl 1246.76018)] to the Navier-Stokes system that involves a probabilistic Lagrangian representation formula making use of stochastic flows. These results are applicable to a more natural class of mild solutions instead of the case of previous works related to classical ones.Enhanced diffusivity in perturbed senile reinforced random walk modelshttps://zbmath.org/1472.352912021-11-25T18:46:10.358925Z"Dinh, Thu"https://zbmath.org/authors/?q=ai:dinh.thu"Xin, Jack"https://zbmath.org/authors/?q=ai:xin.jack-xSummary: We consider diffusivity of random walks with transition probabilities depending on the number of consecutive traversals of the last traversed edge, the so called senile reinforced random walk (SeRW). In one dimension, the walk is known to be sub-diffusive with identity reinforcement function. We perturb the model by introducing a small probability \(\delta\) of escaping the last traversed edge at each step. The perturbed SeRW model is diffusive for any \(\delta>0\), with enhanced diffusivity \((\gg O(\delta^2))\) in the small \(\delta\) regime. We further study stochastically perturbed SeRW models by having the last edge escape probability of the form \(\delta\xi_n\) with \(\xi_n\)'s being independent random variables. Enhanced diffusivity in such models are logarithmically close to the so called residual diffusivity (positive in the zero \(\delta\) limit), with diffusivity between \(O(\frac{1}{|\log\delta|})\) and \(O(\frac{1}{\log |\log\delta|})\). Finally, we generalize our results to higher dimensions where the unperturbed model is already diffusive. The enhanced diffusivity can be as much as \(O(\log^{-2}\delta)\).Transport equations and perturbations of boundary conditionshttps://zbmath.org/1472.353282021-11-25T18:46:10.358925Z"Tyran-Kamińska, Marta"https://zbmath.org/authors/?q=ai:tyran-kaminska.martaSummary: We provide a new perturbation theorem for substochastic semigroups on abstract AL spaces extending Kato's perturbation theorem to nondensely defined operators. We show how it can be applied to piecewise deterministic Markov processes and transport equations with abstract boundary conditions. We give particular examples to illustrate our results.Stochastic nonlinear thermoelastic system coupled sine-Gordon equation driven by jump noisehttps://zbmath.org/1472.353332021-11-25T18:46:10.358925Z"Cheng, Shuilin"https://zbmath.org/authors/?q=ai:cheng.shuilin"Guo, Yantao"https://zbmath.org/authors/?q=ai:guo.yantao"Tang, Yanbin"https://zbmath.org/authors/?q=ai:tang.yanbinSummary: This paper considers a stochastic nonlinear thermoelastic system coupled sine-Gordon equation driven by jump noise. We first prove the existence and uniqueness of strong probabilistic solution of an initial-boundary value problem with homogeneous Dirichlet boundary conditions. Then we give an asymptotic behavior of the solution.Fokker-Plank system for movement of micro-organism population in confined environmenthttps://zbmath.org/1472.353882021-11-25T18:46:10.358925Z"Fu, Jingyi"https://zbmath.org/authors/?q=ai:fu.jingyi"Perthame, Benoit"https://zbmath.org/authors/?q=ai:perthame.benoit"Tang, Min"https://zbmath.org/authors/?q=ai:tang.minSummary: We consider self-propelled particles confined between two parallel plates, moving with a constant velocity while their moving direction changes by rotational diffusion. The probability distribution of such micro-organisms in confined environment is singular because particles accumulate at the boundaries. This leads us to distinguish between the probability distribution densities in the bulk and in the boundaries. They satisfy a degenerate Fokker-Planck system and we propose boundary conditions that take into account the switching between free-moving and boundary-contacting particles. Relative entropy property, a priori estimates and the convergence to an unique steady state are established. The steady states of both the PDE and individual based stochastic models are compared numerically.Strong Feller property of the magnetohydrodynamics system forced by space-time white noisehttps://zbmath.org/1472.353892021-11-25T18:46:10.358925Z"Yamazaki, Kazuo"https://zbmath.org/authors/?q=ai:yamazaki.kazuoInvariant measures and global well posedness for the SQG equationhttps://zbmath.org/1472.353922021-11-25T18:46:10.358925Z"Földes, Juraj"https://zbmath.org/authors/?q=ai:foldes.juraj"Sy, Mouhamadou"https://zbmath.org/authors/?q=ai:sy.mouhamadouSummary: We construct an invariant measure \(\mu\) for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of \(\mu\) are initial conditions of global, unique solutions of SQG that depend continuously on the initial data. In addition, we show that the support of \(\mu\) is infinite dimensional, meaning that it is not locally a subset of any compact set with finite Hausdorff dimension. Also, there are global solutions that have arbitrarily large initial condition. The measure a \(\mu\) is obtained via fluctuation-dissipation method, that is, as a limit of invariant measures for stochastic SQG with a carefully chosen dissipation and random forcing.The Strassen invariance principle for certain non-stationary Markov-Feller chainshttps://zbmath.org/1472.353962021-11-25T18:46:10.358925Z"Czapla, Dawid"https://zbmath.org/authors/?q=ai:czapla.dawid"Horbacz, Katarzyna"https://zbmath.org/authors/?q=ai:horbacz.katarzyna"Wojewódka-Ściążko, Hanna"https://zbmath.org/authors/?q=ai:wojewodka-sciazko.hannaSummary: We propose certain conditions implying the functional law of the iterated logarithm (the Strassen invariance principle) for some general class of non-stationary Markov-Feller chains. This class may be briefly specified by the following two properties: firstly, the transition operator of the chain under consideration enjoys a non-linear Lyapunov-type condition, and secondly, there exists an appropriate Markovian coupling whose transition probability function can be decomposed into two parts, one of which is contractive and dominant in some sense. Our criterion may serve as a useful tool in verifying the functional law of the iterated logarithm for certain random dynamical systems, developed e.g. in biology and population dynamics. In the final part of the paper we present an example application of our main theorem to a mathematical model describing stochastic dynamics of gene expression.The Dirichlet problem for nonlocal elliptic equationshttps://zbmath.org/1472.354222021-11-25T18:46:10.358925Z"Tian, Rongrong"https://zbmath.org/authors/?q=ai:tian.rongrong"Wei, Jinlong"https://zbmath.org/authors/?q=ai:wei.jinlong"Tang, Yanbin"https://zbmath.org/authors/?q=ai:tang.yanbinSummary: We study a class of nonlocal elliptic equations \((-\Delta)^{\alpha/2}\rho(x)-b(x)\cdot\nabla\rho(x)=f(x)\) on a bounded domain \(\Omega\subset\mathbb{R}^d\). For \(f\in L^q(\Omega)(q>d/\alpha)\), by the Lax-Milgram theorem and the De Giorgi iteration, we prove the existence and uniqueness of \(L^\infty\) solution. Furthermore, we investigate the existence of densities for measure-valued solutions to nonhomogeneous measure-valued nonlocal elliptic equations.Corrigendum to: ``Analysis of regularized inversion of data corrupted by white Gaussian noise''https://zbmath.org/1472.354542021-11-25T18:46:10.358925Z"Kekkonen, Hanne"https://zbmath.org/authors/?q=ai:kekkonen.hanne"Lassas, Matti"https://zbmath.org/authors/?q=ai:lassas.matti-j"Siltanen, Samuli"https://zbmath.org/authors/?q=ai:siltanen.samuliCorrigendum to the authors' paper [ibid. 30, No. 4, Article ID 045009, 18 p. (2014; Zbl 1287.35101)].Decomposition of random walk measures on the one-dimensional torushttps://zbmath.org/1472.370072021-11-25T18:46:10.358925Z"Gilat, Tom"https://zbmath.org/authors/?q=ai:gilat.tomSummary: The main result of this paper is a decomposition theorem for a measure on the one-dimensional torus. Given a sufficiently large subset \(S\) of the positive integers, an arbitrary measure on the torus is decomposed as the sum of two measures. The first one \(\mu_1\) has the property that the random walk with initial distribution \(\mu_1\) evolved by the action of \(S\) equidistributes very fast. The second measure \(\mu_2\) in the decomposition is concentrated on very small neighborhoods of a small number of points.Markov random walks on homogeneous spaces and Diophantine approximation on fractalshttps://zbmath.org/1472.370082021-11-25T18:46:10.358925Z"Prohaska, Roland"https://zbmath.org/authors/?q=ai:prohaska.roland"Sert, Cagri"https://zbmath.org/authors/?q=ai:sert.cagriAuthors' abstract: In the first part, using the recent measure classification results of Eskin-Lindenstrauss, we give a criterion to ensure a.s. equidistribution of empirical measures of an i.i.d. random walk on a homogeneous space \(G/\Gamma \). Employing renewal and joint equidistribution arguments, this result is generalized in the second part to random walks with Markovian dependence. Finally, following a strategy of \textit{D. Simmons} and \textit{B. Weiss} [Invent. Math. 216, No. 2, 337--394 (2019; Zbl 1454.22009)], we apply these results to Diophantine approximation problems on fractals and show that almost every point with respect to Hausdorff measure on a graph directed self-similar set is of generic type, so, in particular, well approximable.Dynamical Borel-Cantelli lemmashttps://zbmath.org/1472.370092021-11-25T18:46:10.358925Z"Xing, Viktoria"https://zbmath.org/authors/?q=ai:xing.viktoriaThe classical Borel-Cantelli lemma says that if the sum of the probabilities of a collection of independent events is infinite, then the probability of the occurrence of infinitely many of these events must be one. There is a number of generalizations of this result for dynamical systems, when the assumption about independence cannot take place. In particular, \textit{D. Y. Kleinbock} and \textit{G. A. Margulis} [Invent. Math. 138, No. 3, 451--494 (1999; Zbl 0934.22016)] have given a useful sufficient condition for strongly Borel-Cantelli sequences, based on the work of \textit{W. Schmidt} [Can. J. Math. 12, 619--631 (1960; Zbl 0097.26205)]. The author obtains a weaker sufficient condition for the strongly Borel-Cantelli sequences, which allows to extend known results of Borel-Cantelli lemma type for dependent variables. Applications to one-dimensional Gibbs-Markov systems are established.A Poisson limit theorem for Gibbs-Markov mapshttps://zbmath.org/1472.370102021-11-25T18:46:10.358925Z"Zhang, Xuan"https://zbmath.org/authors/?q=ai:zhang.xuanThe main result of this paper is a Poisson limit theorem for the number of visits of Gibbs-Markov maps to a sequence of shrinking sets with bounded cylindrical lengths. If this sequence converges to a point, this could be either a periodic point or a compactification point. Applications of this result to continued fractions are also given, including a generalization of Poisson limit theorem by \textit{W. Doeblin} [Compos. Math. 7, 353--371 (1940; Zbl 0022.37001)]
for the number of large terms in the continued fraction expansions of irrational numbers in the unit interval.Limiting distribution of geodesics in a geometrically finite quotients of regular treeshttps://zbmath.org/1472.370412021-11-25T18:46:10.358925Z"Kwon, Sanghoon"https://zbmath.org/authors/?q=ai:kwon.sanghoon"Lim, Seonhee"https://zbmath.org/authors/?q=ai:lim.seonheeSummary: Let \(\mathcal{T}\) be a \((q+1)\)-regular tree and let \(\Gamma\) be a geometrically finite discrete subgroup of the group \(\mathrm{Aut}(\mathcal{T})\) of automorphisms of \(\mathcal{T}\). In this article, we prove an extreme value theorem on the distribution of geodesics in a non-compact quotient graph \(\Gamma\backslash\mathcal{T}\). Main examples of such graphs are quotients of a Bruhat-Tits tree by non-cocompact discrete subgroups \(\Gamma\) of \(\mathrm{PGL}(2,\mathbf{K})\) of a local field \(\mathbf{K}\) of positive characteristic.
We investigate, for a given time \(T\), the measure of the set of \(\Gamma\)-equivalent classes of geodesics with distance at most \(N(T)\) from a sufficiently large fixed compact subset \(D\) of \(\Gamma\backslash\mathcal{T}\) up to time \(T\). We show that there exists a function \(N(T)\) such that for Bowen-Margulis measure \(\mu\) on the space \(\Gamma\backslash\mathcal{GT}\) of geodesics and the critical exponent \(\delta\) of \(\Gamma\),
\[
\lim\limits_{T\to\infty}\mu(\{[l]\in\Gamma\backslash\mathcal{GT}:\underset{0\le t \le T}\max d(D,l(t))\le N(T)+y\})=e^{-q^y/e^{2\delta y}}.
\]
In fact, we obtain a precise formula for \(N(T)\): there exists a constant \(C\) depending on \(\Gamma\) and \(D\) such that
\[
N(T)=\log_{e^{2\delta/q}}\Big(\frac{T(e^{2\delta-q)}}{2e^{2\delta}-C(e^{2\delta}-q)}\Big).\]Multivariate Laplace's approximation with estimated error and application to limit theoremshttps://zbmath.org/1472.410162021-11-25T18:46:10.358925Z"Łapiński, Tomasz M."https://zbmath.org/authors/?q=ai:lapinski.tomasz-mThe authors study the asymptotics of the multi-dimensional Laplace integrals \[ I_1(N):=\int_\Omega g(x)e^{Nf(x,N)}dx \] and \[ I_2(N):=\int_{\Omega\cap\{x;x_1\ge 0\}} g(x)e^{Nf(x,N)}dx \] where \(\Omega\subset R^m\) is an open set, the functions \(f\) and \(g\) are sufficiently regular, and \(N\in Z^+\) is a large parameter. In the first integral, \(f(\cdot,N)\) has a unique maximum \(x^*(N)\) in the interior of \(\Omega\), whereas in the second one, \(f(\cdot,N)\) has a unique non-critical maximum on the boundary \(\{x;x_1= 0\}\).
The asymptotic theory of these integrals is well-known, and we can find in the literature different asymptotic expansions for them. The form of these expansions depends on the location and on the order of the critical points of the exponent. The authors go a step further and derive the following error bounds for the first order Laplace approximation:
\noindent (1) For the second integral with \(\Omega=[0,\infty)\) (univariate integral), the authors derive the following formula \[ \int_\Omega g(x)e^{Nf(x,N)}dx=E^{\frac{Nf(0,N)}{N}}\left(\frac{g(0)}{\vert f'(0,N)\vert}+\frac{w(N)}{N}\right), \] with \(w(N)=\mathcal{O}(1)\) as \(N\to\infty\). Moreover, an explicit bound for \(w(N)\) is given in terms of bounds for the first partial derivatives of the phase function \(f\).
\noindent (2) For the first integral, when the maximum \(x^*\) is an interior point, the authors derive the following formula \[ \int_\Omega g(x)e^{Nf(x,N)}dx=E^{Nf(x^*,N)}\left(\frac{2\pi}{N}\right)^{m/2}\left(\frac{g(x^*)}{\sqrt{\vert D^2 f(x^*,N)\vert}}+\frac{w(N)}{\sqrt{N}}\right), \] with \(w(N)=\mathcal{O}(1)\) as \(N\to\infty\) an a similar explicit bound for \(w(N)\).
\noindent (3) For the second integral, when the maximum \(x^*\) is on the boundary, the authors derive the following formula \[ \int_{\Omega\cap\{x;x_1\ge 0\}} g(x)e^{Nf(x,N)}dx=E^{\frac{Nf(x^*,N)}{N}}\left(\frac{2\pi}{N}\right)^{(m-1)/2}\left(\frac{g(x^*)}{\sqrt{\vert D_y^2 f(x^*,N)\vert}}+\frac{w(N)}{\sqrt{N}}\right), \] with \(w(N)=\mathcal{O}(1)\) as \(N\to\infty\) an a similar explicit bound for \(w(N)\). The symbol \(D_y^2f\) represents the Hessian matrix of \(f\) as a function of all its variables except \(x_1\).
As an application, the authors prove the weak law of large numbers and the central limit theorem, with different limit distributions depending on the location of the maximum \(x^*\) of the phase function: when the maximum is in the interior of the integration domain, the limit distribution is a Normal distribution. When the maximum is on the boundary, the limit distribution is exponential in one direction and Normal in the other directions.Multiplier theorems via martingale transformshttps://zbmath.org/1472.420132021-11-25T18:46:10.358925Z"Bañuelos, Rodrigo"https://zbmath.org/authors/?q=ai:banuelos.rodrigo"Baudoin, Fabrice"https://zbmath.org/authors/?q=ai:baudoin.fabrice"Chen, Li"https://zbmath.org/authors/?q=ai:chen.li.2|chen.li.4|chen.li.5|chen.li.6|chen.li.7|chen.li.1|chen.li.3"Sire, Yannick"https://zbmath.org/authors/?q=ai:sire.yannickSummary: We develop a new and general approach to prove multiplier theorems in various geometric settings. The main idea is to use martingale transforms and a Gundy-Varopoulos representation for multipliers defined via a suitable extension procedure. Along the way, we provide a probabilistic proof of a generalization of a result by \textit{P. R. Stinga} and \textit{J. L. Torrea} [Commun. Partial Differ. Equations 35, No. 10--12, 2092--2122 (2010; Zbl 1209.26013)], which is of independent interest. Our methods here also recover the sharp \(L^p\) bounds for second order Riesz transforms by a limiting argument.Hypergroups and distance distributions of random walks on graphshttps://zbmath.org/1472.430092021-11-25T18:46:10.358925Z"Endo, Kenta"https://zbmath.org/authors/?q=ai:endo.kenta"Mimura, Ippei"https://zbmath.org/authors/?q=ai:mimura.ippei"Sawada, Yusuke"https://zbmath.org/authors/?q=ai:sawada.yusukeSummary: Wildberger's construction enables us to obtain a hypergroup from a random walk on a special graph. We will give a probability theoretic interpretation to products on the hypergroup. The hypergroup can be identified with a commutative algebra whose basis is transition matrices. We will estimate the operator norm of such a transition matrix and clarify a relationship between their matrix products and random walks.Martingale inequalities and fractional integral operator in variable Hardy-Lorentz spaceshttps://zbmath.org/1472.460332021-11-25T18:46:10.358925Z"Zeng, Dan"https://zbmath.org/authors/?q=ai:zeng.danSummary: Let \((\Omega, \mathcal{F}, \mathbb{P})\) be a complete probability space. We introduce variable Lorentz space \(\mathcal{L}_{p ( \cdot ) , q}(\Omega)\) defined by rearrangement functions and its related properties. Then, we establish martingale inequalities among these martingale Hardy-Lorentz spaces \(\mathcal{H}_{p ( \cdot ) , q}(\Omega)\) by applying the interpolation theorem. Furthermore, we study the boundedness of the fractional integral operator in variable martingale Hardy spaces \(\mathcal{H}_{p ( \cdot )}^M(\Omega)\) and \(\mathcal{H}_{p ( \cdot ) , q}^M(\Omega)\).Erratum to: ``Orlicz norm and Sobolev-Orlicz capacity on ends of tree based on probabilistic Bessel kernels''https://zbmath.org/1472.460362021-11-25T18:46:10.358925Z"Hara, C."https://zbmath.org/authors/?q=ai:hara.chiaki|hara.carmem"Iijima, R."https://zbmath.org/authors/?q=ai:iijima.ryota"Kaneko, H."https://zbmath.org/authors/?q=ai:kaneko.hiroyuki|kaneko.hiromichi|kaneko.hideaki|kaneko.hajime|kaneko.hiromi|kaneko.hiroshi|kaneko.hiroki|kaneko.haruhiko"Matsumoto, H."https://zbmath.org/authors/?q=ai:matsumoto.hideyuki|matsumoto.hironori|matsumoto.hiroaki|matsumoto.hirotaka|matsumoto.hisayoshi|matsumoto.hideya|matsumoto.hideki|matsumoto.hiroshi|matsumoto.hisaaki|matsumoto.hiroyuki.1From the text: The notation \(\varphi(t)\) in the integrands of the statements of Proposition 6.2, Theorem 6.3, Theorem 6.4 and in the proof of Theorem 6.4 in the authors' paper [ibid. 7, No. 1, 24--38 (2015; Zbl 1341.46023)] must be replaced by \(\varphi(1/t)\).An operad of non-commutative independences defined by treeshttps://zbmath.org/1472.460682021-11-25T18:46:10.358925Z"Jekel, David"https://zbmath.org/authors/?q=ai:jekel.david"Liu, Weihua"https://zbmath.org/authors/?q=ai:liu.weihuaSummary: %\DeclareMathSymbol{\boxright}{3}{mathb}{'151}
We study certain notions of \(N\)-ary non-commutative independence, which generalize free, Boolean, and monotone independence. For every rooted subtree \(\mathcal{T}\) of an \(N\)-regular rooted tree, we define the \(\mathcal{T}\)-free product of \(N\) non-commutative probability spaces and the \(\mathcal{T}\)-free additive convolution of \(N\) non-commutative laws.
These \(N\)-ary additive convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the operadic framework, the proof of convolution identities such as
%\(\mu\boxplus\nu=\mu\rhd ( \nu\boxempty\kern-9pt\vdash \mu ) \)
%\(\mu\boxplus\nu=\mu\rhd(\nu \boxright \mu)\)
\(\mu\boxplus\nu = \mu \vartriangleright (\nu\,\square\!\!\!\!\!\vdash \!\mu) \)
can be reduced to combinatorial manipulations of trees. In particular, we obtain a decomposition of the \(\mathcal{T} \)-free convolution into iterated Boolean and orthogonal convolutions, which generalizes work of \textit{R. Lenczewski} [J. Funct. Anal. 246, No. 2, 330--365 (2007; Zbl 1129.46055)].
We also develop a theory of \(\mathcal{T} \)-free independence that closely parallels the free, Boolean, and monotone cases, provided that the root vertex has more than one neighbor. This includes combinatorial moment formulas, cumulants, a central limit theorem, and classification of distributions that are infinitely divisible with bounded support. In particular, we study the case where the root vertex of \(\mathcal{T}\) has \(n\) children and each other vertex has \(d\) children, and we relate the \(\mathcal{T} \)-free convolution powers to free and Boolean convolution powers and the Belinschi-Nica semigroup.On the support of the free additive convolutionhttps://zbmath.org/1472.460692021-11-25T18:46:10.358925Z"Bao, Zhigang"https://zbmath.org/authors/?q=ai:bao.zhigang"Erdős, László"https://zbmath.org/authors/?q=ai:erdos.laszlo"Schnelli, Kevin"https://zbmath.org/authors/?q=ai:schnelli.kevinSummary: We consider the free additive convolution of two probability measures \(\mu\) and \(\nu\) on the real line and show that \(\mu\boxplus v\) is supported on a single interval if \(\mu\) and \(\nu\) each has single interval support. Moreover, the density of \(\mu\boxplus\nu\) is proven to vanish as a square root near the edges of its support if both \(\mu\) and \(\nu\) have power law behavior with exponents between \(-1\) and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [\textit{Z.-G. Bao} et al., J. Funct. Anal. 279, No. 7, Article ID 108639, 93~p. (2020; Zbl 1460.46058)].Max-convolution semigroups and extreme values in limit theorems for the free multiplicative convolutionhttps://zbmath.org/1472.460702021-11-25T18:46:10.358925Z"Ueda, Yuki"https://zbmath.org/authors/?q=ai:ueda.yukiSummary: We investigate relations between additive convolution semigroups and max-con\-vo\-lu\-tion semigroups through the law of large numbers for the free multiplicative convolution. Based on these relations, we give a formula related with the Belinschi-Nica semigroup and the max-Belinschi-Nica semigroup. Finally, we give several limit theorems for classical, free and Boolean extreme values.An invitation to optimal transport, Wasserstein distances, and gradient flowshttps://zbmath.org/1472.490012021-11-25T18:46:10.358925Z"Figalli, Alessio"https://zbmath.org/authors/?q=ai:figalli.alessio"Glaudo, Federico"https://zbmath.org/authors/?q=ai:glaudo.federicoThis graduate text offers a relatively self-contained introduction to the optimal transport theory. It consists of five chapters and two appendices.
Chapter 1 gives a brief review of the optimal transport theory, recalls certain of basics of measure theory and Riemannian geometry, and shows three typical examples of the transport maps in connection to the classical isoperimetry.
Chapter 2 presents the so-called core of the optimal transport theory: the solution to Kantorovich's problem for general costs; the duality theory; the solution to Monge's problem for suitable costs.
Chapter 3 utilizes the \([1,\infty)\ni p\)-Wasserstein distances to handle an essential relationship among the optimal transport theory, gradient flows in the Hilbert spaces, and partial differential equations.
Chapter 4 shows a differential viewpoint of the optimal transport theory via studying Benamou-Brenier's and Otto's formulas based on the probability measures.
Chapter 5 suggests several applied topics of the optimal transport theory.
Appendix A includes a set of some interesting exercises and their solutions.
Appendix B outlines a proof of the disintegration theorem.The optimal control problem with state constraints for fully coupled forward-backward stochastic systems with jumpshttps://zbmath.org/1472.490392021-11-25T18:46:10.358925Z"Wei, Qingmeng"https://zbmath.org/authors/?q=ai:wei.qingmengSummary: We focus on the fully coupled forward-backward stochastic differential equations with jumps and investigate the associated stochastic optimal control problem (with the nonconvex control and the convex state constraint) along with stochastic maximum principle. To derive the necessary condition (i.e., stochastic maximum principle) for the optimal control, first we transform the fully coupled forward-backward stochastic control system into a fully coupled backward one; then, by using the terminal perturbation method, we obtain the stochastic maximum principle. Finally, we study a linear quadratic model.An optimal control problem of forward-backward stochastic Volterra integral equations with state constraintshttps://zbmath.org/1472.490402021-11-25T18:46:10.358925Z"Wei, Qingmeng"https://zbmath.org/authors/?q=ai:wei.qingmeng"Xiao, Xinling"https://zbmath.org/authors/?q=ai:xiao.xinlingSummary: This paper is devoted to the stochastic optimal control problems for systems governed by forward-backward stochastic Volterra integral equations (FBSVIEs, for short) with state constraints. Using Ekeland's variational principle, we obtain one kind of variational inequalities. Then, by dual method, we derive a stochastic maximum principle which gives the necessary conditions for the optimal controls.A projected primal-dual gradient optimal control method for deep reinforcement learninghttps://zbmath.org/1472.490422021-11-25T18:46:10.358925Z"Gottschalk, Simon"https://zbmath.org/authors/?q=ai:gottschalk.simon"Burger, Michael"https://zbmath.org/authors/?q=ai:burger.michael"Gerdts, Matthias"https://zbmath.org/authors/?q=ai:gerdts.matthiasSummary: In this contribution, we start with a policy-based Reinforcement Learning ansatz using neural networks. The underlying Markov Decision Process consists of a transition probability representing the dynamical system and a policy realized by a neural network mapping the current state to parameters of a distribution. Therefrom, the next control can be sampled. In this setting, the neural network is replaced by an ODE, which is based on a recently discussed interpretation of neural networks. The resulting infinite optimization problem is transformed into an optimization problem similar to the well-known optimal control problems. Afterwards, the necessary optimality conditions are established and from this a new numerical algorithm is derived. The operating principle is shown with two examples. It is applied to a simple example, where a moving point is steered through an obstacle course to a desired end position in a 2D plane. The second example shows the applicability to more complex problems. There, the aim is to control the finger tip of a human arm model with five degrees of freedom and 29 Hill's muscle models to a desired end position.Viscosity solutions for controlled McKean-Vlasov jump-diffusionshttps://zbmath.org/1472.490542021-11-25T18:46:10.358925Z"Burzoni, Matteo"https://zbmath.org/authors/?q=ai:burzoni.matteo"Ignazio, Vincenzo"https://zbmath.org/authors/?q=ai:ignazio.vincenzo"Reppen, A. Max"https://zbmath.org/authors/?q=ai:reppen.a-max"Soner, H. M."https://zbmath.org/authors/?q=ai:soner.halil-meteThe paper deals with a class of nonlinear integro-differential equations on a subspace of all probability measures on the real line related to the optimal control of McKean-Vlasov jump-diffusions.
The authors investigated an intrinsic notion of viscosity solutions that does not rely on the lifting to a Hilbert space and proved a comparison theorem for these solutions.Leaves decompositions in Euclidean spaceshttps://zbmath.org/1472.520052021-11-25T18:46:10.358925Z"Ciosmak, Krzysztof J."https://zbmath.org/authors/?q=ai:ciosmak.krzysztof-jSummary: We partly extend the localisation technique from convex geometry to the multiple constraints setting.
For a given 1-Lipschitz map \(u:\mathbb{R}^n\to\mathbb{R}^m\), \(m\leq n\), we define and prove the existence of a partition of \(\mathbb{R}^n\), up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of \(u\) is an isometry on these sets.
We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension \(m\), the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag.Random Gale diagrams and neighborly polytopes in high dimensionshttps://zbmath.org/1472.520182021-11-25T18:46:10.358925Z"Schneider, Rolf"https://zbmath.org/authors/?q=ai:schneider.rolf-gA convex polytope \(P\) in Euclidean space \(\mathbb{R}^d\) is \(k\)-neighborly if any \(k\) or fewer vertices of \(P\) are neighbors, i.e. if their convex hull is a face of \(P\). The author recalls a suggestion of David Gale from 1956 and generates sets of combinatorially isomorphic polytopes by choosing their Gale diagrams at random. Importantly, the paper provides a definition of a random Gale diagram. Inspired by a result of [\textit{D. L. Donoho} and \textit{J. Tanner}, Proc. Natl. Acad. Sci. USA 102, No. 27, 9452--9457 (2005; Zbl 1135.60300)], Theorem 1 shows that in high dimensions and under suitable assumptions on the growth of several parameters, the obtained random polytopes have strong neighborliness properties with high probability. Theorem 2 considers the expectation of the involved random variables and describes a phase transition with an explicit threshold.Control of connectivity and rigidity in prismatic assemblieshttps://zbmath.org/1472.520282021-11-25T18:46:10.358925Z"Choi, Gary P. T."https://zbmath.org/authors/?q=ai:choi.gary-pui-tung"Chen, Siheng"https://zbmath.org/authors/?q=ai:chen.siheng"Mahadevan, L."https://zbmath.org/authors/?q=ai:mahadevan.lakshminarayananSummary: How can we manipulate the topological connectivity of a three-dimensional prismatic assembly to control the number of internal degrees of freedom and the number of connected components in it? To answer this question in a deterministic setting, we use ideas from elementary number theory to provide a hierarchical deterministic protocol for the control of rigidity and connectivity. We then show that it is possible to also use a stochastic protocol to achieve the same results via a percolation transition. Together, these approaches provide scale-independent algorithms for the cutting or gluing of three-dimensional prismatic assemblies to control their overall connectivity and rigidity.Stability of the cut locus and a central limit theorem for Fréchet means of Riemannian manifoldshttps://zbmath.org/1472.530452021-11-25T18:46:10.358925Z"Eltzner, Benjamin"https://zbmath.org/authors/?q=ai:eltzner.benjamin"Galaz-García, Fernando"https://zbmath.org/authors/?q=ai:galaz-garcia.fernando"Huckemann, Stephan F."https://zbmath.org/authors/?q=ai:huckemann.stephan-f"Tuschmann, Wilderich"https://zbmath.org/authors/?q=ai:tuschmann.wilderichSummary: We obtain a central limit theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin's Omnibus central limit theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the manifold is compact but may not be satisfied in the non-compact case.Points on nodal lines with given directionhttps://zbmath.org/1472.580202021-11-25T18:46:10.358925Z"Rudnick, Zeév"https://zbmath.org/authors/?q=ai:rudnick.zeev"Wigman, Igor"https://zbmath.org/authors/?q=ai:wigman.igorThis paper treats the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. Furthermore, the authors give upper bounds for the flat torus, and a computation of the expected number for arithmetic random waves is executed.
More precisely, let $\Omega$ be a planar domain with piecewise smooth boundary, and let $f$ be an eigenfunction of the Dirichlet Laplacian with eigenvalue $E$ such that $- \varDelta f = E f$. Given a direction $\zeta \in S^1$, let $N_{\zeta}(f)$ be the number of points $x$ on the nodal line $\{ x \in \Omega: \, f(x) = 0 \}$ with normal pointing in the direction $\pm \zeta$, i.e.,
\[
N_{\zeta}(f) := \# \left\{ x \in \Omega: \, f(x) =0, \,\, \frac{ \nabla f(x)}{ \Vert \nabla f(x) \Vert } = \pm \zeta \right\}.
\]
The first result below asserts an upper bound for $N_{\zeta}(f)$ with the only exceptions being when the nodal line contains a closed geodesic. It will follow as a particular case of a structure result on the set
\[
A_{\zeta}(f) := \{ x \in \Omega: \, f(x) =0, \, \langle \nabla f(x), \zeta^{\perp} \rangle = 0 \}
\]
of nodal directional points, i.e., the set of nodal points where $\nabla f$ is orthogonal to $\zeta^{\perp}$, thus it is co-linear to $\zeta$, $A_{\zeta}(f)$ contains all the singular nodal points of $f^{-1}(0)$, and could also contain certain closed geodesics in direction orthogonal to $\zeta$.
Theorem 1. Let $\zeta \in S^1$ be a direction, and $f$ be a toral eigenfunction such that $- \varDelta f = E f$ for some $E > 0$.
(i) If $\zeta$ is rational, then the set $A_{\zeta}(f)$ consists of at most $\sqrt{E}/ \pi h(\zeta)$ closed geodesics orthogonal to $\zeta$, at most $\frac{2}{\pi^2} E$ nonsingular points not lying on the geodesics, and possibly, singular points of the nodal set, where $h(\zeta)$ is the height for a rational vector.
(ii) If $\zeta$ is not rational, then the set $A_{\zeta}(f)$ consists of at most $\frac{2}{\pi^2} E$ nonsingular points, and possibly, singular points of the nodal set.
(iii) In particular, if $A_{\zeta}(f)$ does not contain a closed geodesics, then
\[
N_{\zeta}(f) \leqslant \frac{2}{\pi^2} \cdot E
\]
holds.
Next the authors compute the expected value of $N_{\zeta}$ for arithmetic random waves, cf. [\textit{F. Oravecz} et al., Ann. Inst. Fourier (Grenoble) 58, No.1, 299--335 (2008; Zbl 1153.35058)]. There are random eigenfunctions on the torus,
\[
f(x) = f_n(x) = \sum_{ \lambda \in {\mathcal E}_n } c_{\lambda} \cdot e( \langle \lambda, x \rangle )
\]
where $e(z) = e^{ 2 \pi i z}$ and $ {\mathcal E} := \{ \lambda = ( \lambda_1, \lambda_2) \in{\mathbb Z}: \, \Vert \lambda \Vert^2 = n \}$ is the set of all representations of the integer $n = \lambda_1^2 + \lambda_2^2$ as a sum of two integer squares, and $c_{\lambda}$ are standard Gaussian random variables, identically distributed and independent wave for the constraint $c_{- \lambda} = \bar{ c}_{\lambda}$, making $f_n$ real valued eigenfunctions of the Laplacian with eigenvalue $E = 4 \pi^2 n$ for every choice of the coefficients $\{ a_{\lambda} \}, \lambda \in {\mathcal E}_{\lambda}$. Let $\mu_n$ be the atomic measure on the unit circle given by
\[
\mu_n = \frac{1}{ r_2(n)} \sum_{ \lambda \in {\mathcal E}_n }\delta_{ \lambda / \sqrt{n} },
\]
where $r_2(n) := \# {\mathcal E}_n$, and let
\[
\hat{\mu}_n (k) = \frac{1}{ r_2(n)} \sum_{ \lambda = ( \lambda_1, \lambda_2) \in {\mathcal E}_n } \left( \frac{ \lambda_1 + i \lambda_2}{\sqrt{n} } \right)^k \in {\mathbb R}
\]
be its Fourier coefficients.
Theorem 2. For $\zeta = e^{ i \theta} \in S^1$, the expected value of $N_{\zeta}(f)$ for the arithmetic random wave (1.4) is
\[
{\mathbb E} [ N_{\zeta} ] = \frac{1}{ \sqrt{2} } n ( 1 + \hat{\mu}_n(4) \cdot \cos( 4 \theta) )^{1/2}.
\]
For other related works, see e.g. [\textit{M. Krishnapur} et al., Ann. Math. (2) 177, No. 2, 699--737 (2013; Zbl 1314.60101)] as to nodal length fluctuations for arithmetic random waves, and [\textit{A. Logunov}, Ann. Math. (2) 187, No. 1, 221--239 (2018; Zbl 1384.58020)] for nodal sets of Laplace eigenfunctions.Probability-2. Translated from the fourth Russian edition by R. P. Boas and D. M. Chibisovhttps://zbmath.org/1472.600012021-11-25T18:46:10.358925Z"Shiryaev, Albert N."https://zbmath.org/authors/?q=ai:shiryaev.albert-nThis textbook is the second volume of a pair that presents the latest English edition of the author's classic, Probability. Building on the foundations established in the preceding Probability-1, this volume guides the reader on to the theory of random processes. The new edition includes expanded material on financial mathematics and financial engineering; new problems, exercises, and proofs throughout, and a historical review charting the development of the mathematical theory of probability. Suitable for an advanced undergraduate or beginning graduate student with a course in probability theory, this volume forms the natural sequel to Probability-1 [\textit{A. N. Shiryaev}, Probability-1. Translated from the fourth Russian edition by R. P. Boas and D. M. Chibisov. 3rd edition. New York, NY: Springer (2016; Zbl 1390.60002)].
Probability-2 opens with classical results related to sequences and sums of independent random variables, such as the zero-one laws, convergence of series, strong law of large numbers, and the law of the iterated logarithm. The subsequent chapters go on to develop the theory of random processes with discrete time: stationary processes, martingales, and Markov processes. The historical review illustrates the growth from intuitive notions of randomness in history through to modern day probability theory and theory of random processes.
Along with its companion volume, this textbook presents a systematic treatment of probability from the ground up, starting with intuitive ideas and gradually developing more sophisticated subjects, such as random walks, martingales, Markov chains, the measure-theoretic foundations of probability theory, weak convergence of probability measures, and the central limit theorem. Many examples are discussed in detail, and there are a large number of exercises throughout.
Some new material has also been added to Chapter VII that treats the theory of martingales with discrete time. In Section 9 of that chapter a discrete version of Itô's formula is presented, which may be viewed as an introduction to the stochastic calculus for the Brownian motion, where Itô's (change-of-variables) formula is of key importance. In Section 10, it is shown how the methods of the martingale theory provide a simple way of obtaining estimates of ruin probabilities for an insurance company acting under the Cramer-Lundberg model. The next Section 11 deals with the ``arbitrage theory'' in stochastic financial mathematics. Here he states two ``Fundamental theorems of the arbitrage theory'', which provide conditions in martingale terms for absence of arbitrage possibilities and conditions guaranteeing the existence of a portfolio of assets, which enables one to achieve the objected aim. Finally, Section 13 of that chapter is devoted to the general theory of optimal stopping rules for arbitrary random sequences. The material presented here demonstrates how the concepts and results of the martingale theory can be applied in the various problems of ``Stochastic Optimization''. There is also a number of changes and supplements made in other chapters. There is some new material concerning the theorems on monotone classes (Section 2 of Chapter II), which relies on detailed treatment of the concepts and properties of ``$\pi$-$\lambda$'' systems, and the fundamental theorems of mathematical statistics given in Section 13 of Chapter III.
The novelty of the present edition is also the ``Outline of historical development of the mathematical probability theory'', placed at the end of ``Probability-2''. In a number of sections new problems have been added.Ambit stochasticshttps://zbmath.org/1472.600022021-11-25T18:46:10.358925Z"Barndorff-Nielsen, Ole E."https://zbmath.org/authors/?q=ai:barndorff-nielsen.ole-eiler"Benth, Fred Espen"https://zbmath.org/authors/?q=ai:benth.fred-espen"Veraart, Almut E. D."https://zbmath.org/authors/?q=ai:veraart.almut-e-dThis is a monograph on stochastic modelling of complex phenomena which are random and evolve in both time and space. The word `ambit' used as a part of the title appears for the first time in the literature. Available are only a few `ambit' papers and one weekly `Ambit magazine'. Relying on advanced probability the authors define and intensively use the notions `ambit sets' and `ambit fields'. Important applied problems lead to the necessity to develop a systematic and adequate theory. In order to build up good mathematical models, one needs a general theory of non-semimartingale stochastic integration with respect to Volterra processes followed by a detailed analytical study. The author pays a great attention to diverse methods of numerical integration and simulation algorithms. All these are successfully used to suggest and analyze stochastic models in complex areas such as turbulence and stochastic volatility.
The reader will get a good sense of the contents of the book by looking at the chapter names and, in brackets, the names of two randomly chosen sections. Part I. The purely temporally case: 1. Volatility modulated Volterra processes (Lévy processes, semimartingale and non-semimartingale settings). 2. Simulation (a stepwise simulation scheme based on the Laplace representation, simulation based on numerically solving stochastic PDEs). 3. Asymptotic theory for power variation of LSS processes (convergence concept, Asymptotic theory in the non-semimartingale setting). 4. Integration with respect to volatility modulated Volterra processes (integration with respect to VMBV processes, discussion of stochastic integration based on an infinite dimensional approach). Part II. The spatio-temporal case: 5. The ambit framework (integration concepts with respect to a Lévy basis, general aspects of the theory of ambit fields and processes). 6. Representation and simulation of ambit fields (Fourier transformation of ambit fields, representations of ambit fields in Hilbert space). 7. Stochastic integration with ambit fields as integrators (definition of the stochastic integral, relationships to semimartingale integration). 8. Trawl processes (choices for the marginal distribution, inference for trawl processes). Part III. Applications: 9. Turbulence modelling (exponentiated ambit fields and correlators, some remarks on dynamic intermittency). 10. Stochastic modelling on energy spot prices by LSS processes (case study: electricity spot prices from the European energy exchange market, pricing electricity derivatives). 11. Forward curve modelling by ambit fields (properties of the ambit model, application to spread options). Two appendices: A. Bessel functions. B. Generalized hyperbolic distribution. A comprehensive list of References and Index.
The authors have written a fundamental book on contemporary probability theory and its applications. The book can be strongly recommended to theorists and applied scientists.Topics in infinitely divisible distributions and Lévy processeshttps://zbmath.org/1472.600032021-11-25T18:46:10.358925Z"Rocha-Arteaga, Alfonso"https://zbmath.org/authors/?q=ai:rocha-arteaga.alfonso"Sato, Ken-iti"https://zbmath.org/authors/?q=ai:sato.ken-itiThis is a compact and comprehensive text dealing with the class of \textit{infinitely divisible} distributions and the subclasses of \textit{selfdecomposable} and \textit{stable} distributions. The role of all these is vital when defining and studying \textit{Lévy processes}.
The material is distributed into five chapters. The text is written in good style by giving exact definitions followed by statements and their proofs. The notions and the results are well illustrated by a reasonable number of examples.
It is useful to see the list of the five chapter names and give in brackets key words and phrases showing the contents: 1. Classes \(L_m\) and their characterization (limit theorems, characterization by Lévy-Khintchine representation). 2. Classes \(L_m\) and Ornstein-Uhlenbeck type processes (stochastic integrals based on Lévy processes). 3. Selfsimilar additive processes and stationary Ornstein-Uhlenbeck type processes (Lamperti transformation). 4. Multivariate subordination (subordinators, subordination of cone-parameter Lévy processes). 5. Inheritance in multivariate subordination (strict stability, operator generalization).
Each chapter ends with extremely useful `Notes'. Thus there are five well-written essays containing historical facts and going through important steps in developing the infinite divisibility and the theory of Lévy processes. All comments are supported by referring to original sources.
There is one and half page of `Notations', Bibliography and Index.
This book will be beneficial to the huge audience, from graduate students to active researchers, involved into the area of Lévy processes and their applications.An observation-driven time-dependent basis for a reduced description of transient stochastic systemshttps://zbmath.org/1472.600042021-11-25T18:46:10.358925Z"Babaee, H."https://zbmath.org/authors/?q=ai:babaee.hessamSummary: We present a variational principle for the extraction of a time-dependent orthonormal basis from random realizations of transient systems. The optimality condition of the variational principle leads to a closed-form evolution equation for the orthonormal basis and its coefficients. The extracted modes are associated with the most transient subspace of the system, and they provide a reduced description of the transient dynamics that may be used for reduced-order modelling, filtering and prediction. The presented method is matrix free; it relies only on the observables of the system and ignores any information about the underlying system. In that sense, the presented reduction is purely observation driven and may be applied to systems whose models are not known. The presented method has linear computational complexity and memory storage requirement with respect to the number of observables and the number of random realizations. Therefore, it may be used for a large number of observations and samples. The effectiveness of the proposed method is tested on four examples: (i) stochastic advection equation, (ii) stochastic Burgers equation, (iii) a reduced description of transient instability of Kuramoto-Sivashinsky, and (iv) a transient vertical jet governed by the incompressible Navier-Stokes equation. In these examples, we contrast the performance of the time-dependent basis versus static basis such as proper orthogonal decomposition, dynamic mode decomposition and polynomial chaos expansion.Comparison of variance concepts interpreted by two measurement theorieshttps://zbmath.org/1472.600052021-11-25T18:46:10.358925Z"Ye, Xiaoming"https://zbmath.org/authors/?q=ai:ye.xiaoming"Ding, Shijun"https://zbmath.org/authors/?q=ai:ding.shijunSummary: In several previously published literatures from the author, the concept of variance is proposed as the evaluation of probability interval of an error, instead of as the dispersion of measured value defined by existing measurement theory. In this paper, by comparing the formation process of the two interpretations of variance concept, the author will demonstrate the existing variance concept actually violates a basic mathematical concept that the variance of a definite value is 0, expose where this wrong concept comes from, and then provide proof of the new variance concept.The circular law for sparse non-Hermitian matriceshttps://zbmath.org/1472.600062021-11-25T18:46:10.358925Z"Basak, Anirban"https://zbmath.org/authors/?q=ai:basak.anirban"Rudelson, Mark"https://zbmath.org/authors/?q=ai:rudelson.markLet \(\lambda_1\),\dots, \(\lambda_n\) the eigenvalues of a \(n\times n\) matrix \(B\); its empirical spectral distribution (ESD) is defined by \(L_B:=\frac{1}{n}\,\sum_{i=1}^n \delta_{\lambda_i}\), where \(\delta_x\) is the Dirac measure concentrated at \(x\). The sub-Gausssian norm of a random variable \(\xi\) is defined by
\[
\|\xi\|_{\psi_2}:=\sup_{k\ge 1} k^{-1/2}\,\mathbb{E}^{1/k}(|\xi|^k)\,.
\]
The main result of this important paper is the following theorem, which extends previous results quoted in the introduction.
Theorem. Let \(A_n\) be an \(n\times n\) matrix with i.i.d. entries \(a_{i,j}=\delta_{i,j}\,\xi_{i,j}\), where the \(\delta_{i,j}\) are independent Bernoulli random variables taking the value \(1\) with probability \(p_n\in\left]0,1\right]\) and \(\xi_{i,j}\) are real-valued i.i.d. sub-Gaussian centred random variables with unit variance.
\begin{enumerate}
\item[(i)] If \(p_n\) is such that \(np_n=\omega(\log^2 n)\), then as \(n\to\infty\) the \textnormal{ESD} of \(A_n/\sqrt{n\,p_n}\) converges weakly in probability to the circular law.
\item[(ii)] There exists a constant \(c\), which depends only on the sub-Gaussian norm of \(\{\xi_{i,j}\}\), such that if \(p_n\) satisfies the inequality \(np_n>\exp(c\,\sqrt{\log n})\), then the conclusion of (i) holds almost surely.
\end{enumerate}
Here, if \((a_n)\) and \((b_n)\) are two sequences of positive reals, one writes \(a_n=\omega(b_n)\) if \(b_n=o(a_n)\), \(a_n=O(b_n)\) and \(\limsup_{n\to\infty} a_n/b_n<\infty\)Remarks connected with the weak limit of iterates of some random-valued functions and iterative functional equationshttps://zbmath.org/1472.600072021-11-25T18:46:10.358925Z"Baron, Karol"https://zbmath.org/authors/?q=ai:baron.karolSummary: The paper consists of two parts. At first, assuming that \((\Omega, \mathcal{A}, P)\) is a probability space and \((X, \varrho)\) is a complete and separable metric space with the \(\sigma\)-algebra \(\mathcal{B}\) of all its Borel subsets we consider the set \(\mathcal{R}_c\) of all \(\mathcal{B} \otimes \mathcal{A}\)-measurable and contractive in mean functions \(f: X \times \Omega \rightarrow X\) with finite integral \(\int_\Omega \varrho (f(x, \omega), x) P (d \omega)\) for \(x \in X\), the weak limit \(\pi^f\) of the sequence of \textit{iterates} of \(f \in \mathcal{R}_c\), and investigate continuity-like property of the function \(f \mapsto \pi^f\), \(f \in \mathcal{R}_c\), and Lipschitz solutions \(\varphi\) that take values in a separable Banach space of the equation:
\[
\varphi (x) = \int_\Omega \varphi (f(x,\omega) P ( d\omega) + F(x).
\]
Next, assuming that \(X\) is a real separable Hilbert space, \( \Lambda \): \(X \rightarrow X\) is linear and continuous with \(\Vert \Lambda \Vert < 1\), and \(\mu\) is a probability Borel measure on \(X\) with finite first moment we examine continuous at zero solutions \(\varphi : X \rightarrow \mathbb{C}\) of the equation
\[
\varphi(x) = \hat{\mu}(x)\varphi (\Lambda x)
\]
which characterizes the limit distribution \(\pi^{f}\) for some special \(f \in \mathcal{R}_c\).Traces of powers of matrices over finite fieldshttps://zbmath.org/1472.600082021-11-25T18:46:10.358925Z"Gorodetsky, Ofir"https://zbmath.org/authors/?q=ai:gorodetsky.ofir"Rodgers, Brad"https://zbmath.org/authors/?q=ai:rodgers.bradThe authors consider a prime power \(q=p^r,\) a matrix \(M\) chosen uniformly from the finite unitary group \(\mathrm{U}(n,q)\subset \mathrm{GL}(n,q^2),\) and the sequence \(\{M^i\}_{1\leq i \leq k}\) where \(i\) is not multiple of \(p.\) They prove that the traces of powers of matrices converge to independent uniform variables in \(\mathbb F_{q^2}\) as \(n \rightarrow \infty.\) The rate of convergence is shown to be exponential in \(n^2.\) \newline The related problem of the rate at which characteristic polynomial of \(M\) equidistributes in `short intervals' of \(\mathbb F_{q^2} [T]\) is also considered. \newline Analogous results are also proven for the general linear, special linear, symplectic and orthogonal groups over a finite field. \newline The proofs depend upon applying techniques from analytic number theory over function fields to formulas due to Fulman and others for the probability that the characteristic polynomial of a random matrix equals a given polynomial.Local laws for non-Hermitian random matrices and their productshttps://zbmath.org/1472.600092021-11-25T18:46:10.358925Z"Götze, Friedrich"https://zbmath.org/authors/?q=ai:gotze.friedrich-w"Naumov, Alexey"https://zbmath.org/authors/?q=ai:naumov.a-a"Tikhomirov, Alexander"https://zbmath.org/authors/?q=ai:tikhomirov.alexander-nPartial generalized four moment theorem revisitedhttps://zbmath.org/1472.600102021-11-25T18:46:10.358925Z"Jiang, Dandan"https://zbmath.org/authors/?q=ai:jiang.dandan"Bai, Zhidong"https://zbmath.org/authors/?q=ai:bai.zhi-dongSummary: This is a complementary proof of partial generalized 4 moment theorem (PG4MT) mentioned and described by the authors [ibid. 27, No. 1, 274--294 (2021; Zbl 07282851)]. Since the G4MT proposed in that paper requires both the matrices \(\mathbf{X}\) and \(\mathbf{Y}\) satisfying the assumption \(\max_{t,s}|u_{ts}|^2\mathrm{E}\{|x_{11}|^4I(|x_{11}|<\sqrt{n})-\mu\}\to 0\) with the same \(\mu\) which maybe restrictive in real applications, we proposed a new G4MT, called PG4MT, without proof. After the manuscript posed in ArXiv, the authors received high interests in the proof of PG4MT through private communications and find the PG4MT more general than G4MT, it is necessary to give a detailed proof of it. Moreover, it is found that the PG4MT derives a CLT of spiked eigenvalues of sample covariance matrices which covers the work in [the first author and \textit{J. Yao}, J. Multivariate Anal. 106, 167--177 (2012; Zbl 1301.62049)] as a special case.The smallest eigenvalue distribution of the Jacobi unitary ensembleshttps://zbmath.org/1472.600112021-11-25T18:46:10.358925Z"Lyu, Shulin"https://zbmath.org/authors/?q=ai:lyu.shulin"Chen, Yang"https://zbmath.org/authors/?q=ai:chen.yang.1Summary: In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight \(x^{\alpha}(1 - x)^{\beta}, x \in [0, 1], \alpha, \beta > -1\), the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval \([t, 1]\) is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of this Bessel kernel determinant. A specialization of the results gives the constant in the asymptotics of the probability that the interval \((- a, a), a > 0\) is free of eigenvalues in the Jacobi unitary ensemble with the symmetric weight \((1 - x^2)^{\beta}, x \in [- 1, 1]\).Large deviations for extreme eigenvalues of deformed Wigner random matriceshttps://zbmath.org/1472.600122021-11-25T18:46:10.358925Z"Mckenna, Benjamin"https://zbmath.org/authors/?q=ai:mckenna.benjaminThe purpose of the paper is to prove a large deviation principle (LDP) for the largest eigenvalue of the random matrix \({X_N} = \frac{{{W_N}}}{{\sqrt N }} + {D_N}\), where \(\frac{{{W_N}}}{{\sqrt N }}\) lies in a particular class of real or complex Wigner matrices. In particular this includes Gaussian ensembles with full-rank general deformation. For the non-Gaussian ensembles, the deformation should be diagonal, and the laws of the entries of \({W_N}\) are supposed to have sharp sub-Gaussian Laplace transforms and satisfy certain concentration properties. It is also assumed that \({D_N}\) is a deterministic matrix whose empirical spectral measure tends to a deterministic limit \({\mu _D}\) and whose extreme eigenvalues tend to the edges of \({\mu _D}\). For these ensembles the paper establishes LDP in a restricted range \(( - \infty ,{x_c})\), where \({x_c}\) depends on the deformation only and can be infinite.On the asymptotic normality in a scheme of allocation of non-equiprobable particle setshttps://zbmath.org/1472.600132021-11-25T18:46:10.358925Z"Chistyakov, V. P."https://zbmath.org/authors/?q=ai:chistyakov.vladimir-pSummary: A scheme of allocation of non-equiprobable \(m\)-dependent particle sets into cells is considered. The number of cells and the number of sets of particles are supposed to tend to \(\infty\) and to have the same order of growth. Asymptotic formulas for mathematical expectation and variance of the number of empty cells are obtained along with the sufficient conditions for their asymptotic normality.A multivariate Poisson theorem for the number of solutions of random inclusions close to given vectorshttps://zbmath.org/1472.600142021-11-25T18:46:10.358925Z"Kopytsev, V. A."https://zbmath.org/authors/?q=ai:kopytsev.v-aSummary: We consider the number of solutions of random inclusion over a finite field that differ from a reference vector by no more than a specified number of coordinates. We find conditions on the growth of vector dimensions under which the number of solutions close to some reference vectors are asymptotically independent and their distributions converge to the Poisson distributions.Explicit accuracy estimates for the Poisson approximation to the distribution of the number of solutions of random inclusionshttps://zbmath.org/1472.600152021-11-25T18:46:10.358925Z"Kopytsev, V. A."https://zbmath.org/authors/?q=ai:kopytsev.v-a"Mikhaĭlov, V. G."https://zbmath.org/authors/?q=ai:mikhajlov.v-gSummary: We study the accuracy of the Poisson approximation for the distribution of the number of solutions of the system of random inclusions belonging to a given set of pairwise nonproportional vectors over the finite field. Numerical examples are given.The stability of sets of solutions for systems of equations with random distortionshttps://zbmath.org/1472.600162021-11-25T18:46:10.358925Z"Mikhailov, V. G."https://zbmath.org/authors/?q=ai:mikhajlov.v-g"Volgin, A. V."https://zbmath.org/authors/?q=ai:volgin.artem-vSummary: Let in a system of equations left sides be functions from \(\{0,1,\dots,N-1\}\) to \(\{0,1\}\) and in the other system all equations be obtained from the equations of the first one by random distortions of the truth tables of these functions. We find conditions on the probability laws of distortions under which the set of solutions of the second system includes all, some or no solutions of the first system.Liouville quantum gravity surfaces with boundary as matings of treeshttps://zbmath.org/1472.600172021-11-25T18:46:10.358925Z"Ang, Morris"https://zbmath.org/authors/?q=ai:ang.morris"Gwynne, Ewain"https://zbmath.org/authors/?q=ai:gwynne.ewainSummary: For \(\gamma\in (0,2)\), the quantum disk and \(\gamma\)-quantum wedge are two of the most natural types of Liouville quantum gravity (LQG) surfaces with boundary. These surfaces arise as scaling limits of finite and infinite random planar maps with boundary, respectively. We show that the left/right quantum boundary length process of a space-filling \(\mathrm{SLE}_{16/\gamma^2}\) curve on a quantum disk or on a \(\gamma\)-quantum wedge is a certain explicit conditioned two-dimensional Brownian motion with correlation \(-\cos(\pi\gamma^2/4)\). This extends the mating of trees theorem of Duplantier, Miller, and Sheffield [\textit{B. Duplantier} et al., ``Liouville quantum gravity as a mating of trees'', Preprint, \url{arXiv:1409.7055}] to the case of quantum surfaces with boundary (the disk case for \(\gamma\in(\sqrt{2},2)\) was previously treated by Duplantier, Miller, Sheffield using different methods). As an application, we give an explicit formula for the conditional law of the LQG area of a quantum disk given its boundary length by computing the law of the corresponding functional of the correlated Brownian motion.Geometry of uniform spanning forest components in high dimensionshttps://zbmath.org/1472.600182021-11-25T18:46:10.358925Z"Barlow, Martin T."https://zbmath.org/authors/?q=ai:barlow.martin-t"Járai, Antal A."https://zbmath.org/authors/?q=ai:jarai.antal-aSummary: We study the geometry of the component of the origin in the uniform spanning forest of \(\mathbb{Z}^d\) and give bounds on the size of balls in the intrinsic metric.Sharp phase transition for the continuum Widom-Rowlinson modelhttps://zbmath.org/1472.600192021-11-25T18:46:10.358925Z"Dereudre, David"https://zbmath.org/authors/?q=ai:dereudre.david"Houdebert, Pierre"https://zbmath.org/authors/?q=ai:houdebert.pierreSummary: The Widom-Rowlinson model (or the Area-interaction model) is a Gibbs point process in \(\mathbb{R}^d\) with the formal Hamiltonian defined as the volume of \(\cup_{x\in\omega}B_1(x)\), where \(\omega\) is a locally finite configuration of points and \(B_1(x)\) denotes the unit closed ball centred at \(x\). The model is also tuned by two other parameters: the activity \(z>0\) related to the intensity of the process and the inverse temperature \(\beta\geq 0\) related to the strength of the interaction. In the present paper we investigate the phase transition of the model in the point of view of percolation theory and the liquid-gas transition. First, considering the graph connecting points with distance smaller than \(2r>0\), we show that for any \(\beta\geq 0\), there exists \(0<\widetilde{z}_c^a(\beta ,r)<+\infty\) such that an exponential decay of connectivity at distance \(n\) occurs in the subcritical phase (i.e. \(z<\widetilde{z}_c^a(\beta ,r))\) and a linear lower bound of the connection at infinity holds in the supercritical case (i.e. \(z>\widetilde{z}_c^a(\beta,r))\). These results are in the spirit of recent works using the theory of randomised tree algorithms [\textit{H. Duminil-Copin} et al., Probab. Theory Relat. Fields 173, No. 1--2, 479--490 (2019; Zbl 07030876); Ann. Math. (2) 189, No. 1, 75--99 (2019; Zbl 07003145)]. Secondly we study a standard liquid-gas phase transition related to the uniqueness/non-uniqueness of Gibbs states depending on the parameters \(z,\beta\). Old results [\textit{D. Ruelle}, ``Existence of a phase transition in a continuous classical system'', Phys. Rev. Lett. 27, 1040--1041 (1971; \url{doi:10.1103/PhysRevLett.27.1040}; \textit{A. Mazel} et al., J. Stat. Phys. 159, No. 5, 1040--1086 (2015; Zbl 1329.82013)] claim that a non-uniqueness regime occurs for \(z=\beta\) large enough and it is conjectured that the uniqueness should hold outside such an half line \((z=\beta\geq\beta_c>0)\). We solve partially this conjecture in any dimension by showing that for \(\beta\) large enough the non-uniqueness holds if and only if \(z=\beta\). We show also that this critical value \(z=\beta\) corresponds to the percolation threshold \(\widetilde{z}_c^a(\beta,r)=\beta\) for \(\beta\) large enough, providing a straight connection between these two notions of phase transition.Conformal covariance of the Liouville quantum gravity metric for \(\gamma\in (0,2)\)https://zbmath.org/1472.600202021-11-25T18:46:10.358925Z"Gwynne, Ewain"https://zbmath.org/authors/?q=ai:gwynne.ewain"Miller, Jason"https://zbmath.org/authors/?q=ai:miller.jason-pSummary: For \(\gamma\in (0,2)\), \(U\subset C\), and an instance \(h\) of the Gaussian free field (GFF) on \(U\), the \(\gamma\)-Liouville quantum gravity (LQG) surface associated with \((U,h)\) is formally described by the Riemannian metric tensor \(e^{\gamma h}(dx^2+dy^2)\) on \(U\). Previous work by the authors showed that one can define a canonical metric (distance function) \(D_h\) on \(U\) associated with a \(\gamma\)-LQG surface. We show that this metric is conformally covariant in the sense that it respects the coordinate change formula for \(\gamma\)-LQG surfaces. That is, if \(U\), \(\widetilde{U}\) are domains, \(\phi :U\to\widetilde{U}\) is a conformal transformation, \(Q=2/\gamma+\gamma 2\), and \(\widetilde{h}=h\circ\phi^{-1}+Q\log |(\phi^{-1})^{\prime}|\), then \(D_h(z,w)=D_{\widetilde{h}}(\phi(z),\phi(w))\) for all \(z,w\in U\). This proves that \(D_h\) is intrinsic to the quantum surface structure of \((U,h)\), i.e., it does not depend on the particular choice of parameterization.Fractional extreme distributionshttps://zbmath.org/1472.600212021-11-25T18:46:10.358925Z"Boudabsa, Lotfi"https://zbmath.org/authors/?q=ai:boudabsa.lotfi"Simon, Thomas"https://zbmath.org/authors/?q=ai:simon.thomas"Vallois, Pierre"https://zbmath.org/authors/?q=ai:vallois.pierreSummary: We consider three classes of linear differential equations on distribution functions, with a fractional order \(\alpha \in [0,1]\). The integer case \(\alpha =1\) corresponds to the three classical extreme families. In general, we show that there is a unique distribution function solving these equations, whose underlying random variable is expressed in terms of an exponential random variable and an integral transform of an independent \(\alpha \)-stable subordinator. From the analytical viewpoint, this distribution is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Fréchet cases, and with a Le Roy function for the Gumbel case.The entropic log odds as a measure of distribution asymmetryhttps://zbmath.org/1472.600222021-11-25T18:46:10.358925Z"Bowden, Roger J."https://zbmath.org/authors/?q=ai:bowden.roger-jSummary: The entropic measure of distributional spread, representing the total binary code complexity of the probability distribution, is equal to the area under the partition entropy function. A conjugate dual, obtained as a simple internal sign change from the partition entropy function, can be employed to indicate the relative complexity to the right of any given point versus that to the left. The area beneath this function constitutes a dual measure of distribution asymmetry, which can be reconciled with one based on conditional expected values.The accumulative law and its probability model: an extension of the Pareto distribution and the log-normal distributionhttps://zbmath.org/1472.600232021-11-25T18:46:10.358925Z"Feng, Minyu"https://zbmath.org/authors/?q=ai:feng.minyu"Deng, Liang-Jian"https://zbmath.org/authors/?q=ai:deng.liangjian"Chen, Feng"https://zbmath.org/authors/?q=ai:chen.feng|chen.feng.1"Perc, Matjaž"https://zbmath.org/authors/?q=ai:perc.matjaz"Kurths, Jürgen"https://zbmath.org/authors/?q=ai:kurths.jurgenSummary: The divergence between the Pareto distribution and the log-normal distribution has been observed persistently over the past couple of decades in complex network research, economics, and social sciences. To address this, we here propose an approach termed as the accumulative law and its related probability model. We show that the resulting accumulative distribution has properties that are akin to both the Pareto distribution and the log-normal distribution, which leads to a broad range of applications in modelling and fitting real data. We present all the details of the accumulative law, describe the properties of the distribution, as well as the allocation and the accumulation of variables. We also show how the proposed accumulative law can be applied to generate complex networks, to describe the accumulation of personal wealth, and to explain the scaling of internet traffic across different domains.The exponentiated discrete inverse Rayleigh distributionhttps://zbmath.org/1472.600242021-11-25T18:46:10.358925Z"Hamed Mashhadzadeh, Zahra"https://zbmath.org/authors/?q=ai:hamed-mashhadzadeh.zahra"Mirmostafaee, S. M. T. K."https://zbmath.org/authors/?q=ai:mirmostafaee.s-m-t-kSummary: In this paper, a new distribution called the exponentiated discrete inverse Rayleigh distribution is introduced, which is an extension of the discrete inverse Rayleigh distribution. This new discrete distribution is a discrete analogue of the continuous exponentiated inverse Rayleigh distribution. In this paper, we discuss the shapes of probability mass and hazard rate functions, the moments of the new distribution and data generation. The maximum likelihood estimation of the parameters is also studied. Finally, an example is given to demonstrate an application of the new distribution.A generalization of the new Weibull Pareto distributionhttps://zbmath.org/1472.600252021-11-25T18:46:10.358925Z"Ibrahim Al-Omari, Amer"https://zbmath.org/authors/?q=ai:ibrahim-al-omari.amer"Al-Khazaleh, Ahmad M. H."https://zbmath.org/authors/?q=ai:al-khazaleh.ahmad-m-h"Alzoubi, Loai M."https://zbmath.org/authors/?q=ai:alzoubi.loai-mSummary: In this paper, a transmuted new Weibull Pareto distribution (NWPD) is suggested as a generalization of the new Weibull Pareto distribution. Some mathematical properties of the transmuted NWP distribution are derived, namely; the moments, failure rate and mean residual life functions, order statistics. Also, the maximum likelihood estimators for the transmuted NWP distribution parameters are provided and its Renyi entropy is proved.Modelling multivariate, overdispersed count data with correlated and non-normal heterogeneity effectshttps://zbmath.org/1472.600262021-11-25T18:46:10.358925Z"Kazemi, Iraj"https://zbmath.org/authors/?q=ai:kazemi.iraj"Hassanzadeh, Fatemeh"https://zbmath.org/authors/?q=ai:hassanzadeh.fatemehSummary: Mixed Poisson models are most relevant to the analysis of longitudinal count data in various disciplines. A conventional specification of such models relies on the normality of unobserved heterogeneity effects. In practice, such an assumption may be invalid, and non-normal cases are appealing. In this paper, we propose a modelling strategy by allowing the vector of effects to follow the multivariate skew-normal distribution. It can produce dependence between the correlated longitudinal counts by imposing several structures of mixing priors. In a Bayesian setting, the estimation process proceeds by sampling variants from the posterior distributions. We highlight the usefulness of our approach by conducting a simulation study and analysing two real-life data sets taken from the German Socioeconomic Panel and the US Centers for Disease Control and Prevention. By a comparative study, we indicate that the new approach can produce more reliable results compared to traditional mixed models to fit correlated count data.Use of the Lévy distribution to adjust data with asymmetry and extreme valueshttps://zbmath.org/1472.600272021-11-25T18:46:10.358925Z"Martínez Naranjo, Jessica Lizeth"https://zbmath.org/authors/?q=ai:martinez-naranjo.jessica-lizeth"Alvear Rodríguez, Carlos Armando"https://zbmath.org/authors/?q=ai:alvear-rodriguez.carlos-armando"Tovar Cueva, José Rafael"https://zbmath.org/authors/?q=ai:tovar-cueva.jose-rafaelSummary: In order to propose a statistical methodology that allows to model asymmetric data using the Lévy distribution, a simulation study is presented under nine different scenarios to evaluate the estimation of the parameters of the distribution in the two approaches of statistics (Classical and Bayesian). The probability distributions Log-Normal, Lévy and Lévy Standard were considered to model the behavior of two real data sets with positive asymmetry, finding that the Lévy distribution fitted well to the proposed data set, therefore the Lévy distribution can be considered as a candidate to adjust asymmetric data with the presence of extreme values.A generalized linear model for multivariate eventshttps://zbmath.org/1472.600282021-11-25T18:46:10.358925Z"Zuniga, Francesco"https://zbmath.org/authors/?q=ai:zuniga.francesco"Kozubowski, Tomasz J."https://zbmath.org/authors/?q=ai:kozubowski.tomasz-j"Panorska, Anna K."https://zbmath.org/authors/?q=ai:panorska.anna-kSummary: We propose a new generalized linear model for modeling multivariate events \((N,X,Y)\), where \(N\) is the duration, \(X\) is the magnitude, and \(Y\) is the maximum of the event. Such events arise, for example, when a process is observed above or below a threshold. Examples include heat waves, flood, draught, or market growth or decline periods. The model is flexible to include different covariates for different parameters. In addition, we propose a new method for checking the goodness of fit (validation) of the model to data. Our goodness of fit methods are based on distributional fit of appropriately transformed data. We include a data example from finance to illustrate the modeling potential of this new generalized linear model.Stein's method for asymmetric \(\alpha \)-stable distributions, with application to the stable CLThttps://zbmath.org/1472.600292021-11-25T18:46:10.358925Z"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng"Nourdin, Ivan"https://zbmath.org/authors/?q=ai:nourdin.ivan"Xu, Lihu"https://zbmath.org/authors/?q=ai:xu.lihuSummary: This paper is concerned with the Stein's method associated with a (possibly) asymmetric \(\alpha \)-stable distribution \(Z\), in dimension one. More precisely, its goal is twofold. In the first part, we exhibit a bound for the Wasserstein distance between \(Z\) and any integrable random variable \(\xi \), in terms of an operator that reduces to the classical fractional Laplacian in the symmetric case. Then, in the second part we apply the aforementioned bound to compute error rates in the stable central limit theorem, when the entries are in the domain \({\mathcal{D}}_\alpha\) of normal attraction of a stable law of exponent \(\alpha \). To conclude, we study the specific case where the entries are Pareto-like multiplied by a slowly varying function, which provides an example of random variables that do not belong to \({\mathcal{D}}_\alpha\) but for which our approach continues to apply.Heavy-tailed distributions, correlations, kurtosis and Taylor's law of fluctuation scalinghttps://zbmath.org/1472.600302021-11-25T18:46:10.358925Z"Cohen, Joel E."https://zbmath.org/authors/?q=ai:cohen.joel-e"Davis, Richard A."https://zbmath.org/authors/?q=ai:davis.richard-a"Samorodnitsky, Gennady"https://zbmath.org/authors/?q=ai:samorodnitsky.gennady-pSummary: \textit{N. S. Pillai} and \textit{X.-L. Meng} [Ann. Stat. 44, No. 5, 2089--2097 (2016; Zbl 1349.62036), p. 2091] speculated that `the dependence among [random variables, rvs] can be overwhelmed by the heaviness of their marginal tails\dots'. We give examples of statistical models that support this speculation. While under natural conditions the sample correlation of regularly varying (RV) rvs converges to a generally random limit, this limit is zero when the rvs are the reciprocals of powers greater than one of arbitrarily (but imperfectly) positively or negatively correlated normals. Surprisingly, the sample correlation of these RV rvs multiplied by the sample size has a limiting distribution on the negative half-line. We show that the asymptotic scaling of Taylor's Law (a power-law variance function) for RV rvs is, up to a constant, the same for independent and identically distributed observations as for reciprocals of powers greater than one of arbitrarily (but imperfectly) positively correlated normals, whether those powers are the same or different. The correlations and heterogeneity do not affect the asymptotic scaling. We analyse the sample kurtosis of heavy-tailed data similarly. We show that the least-squares estimator of the slope in a linear model with heavy-tailed predictor and noise unexpectedly converges much faster than when they have finite variances.On a relation between classical and free infinitely divisible transformshttps://zbmath.org/1472.600312021-11-25T18:46:10.358925Z"Jurek, Zbigniew J."https://zbmath.org/authors/?q=ai:jurek.zbigniew-jSummary: We study two ways (two levels) of finding free-probability analogues of classical infinitely divisible measures. More precisely, we identify their Voiculescu transforms on the imaginary axis. For free-selfdecomposable measures we find a formula (a differential equation) for their background driving transforms. It is different from the one known for classical selfdecomposable measures. We illustrate our methods on hyperbolic characteristic functions. Our approach may produce new formulas for definite integrals of some special functions.Multi operator-stable random measures and fieldshttps://zbmath.org/1472.600322021-11-25T18:46:10.358925Z"Kremer, Dustin"https://zbmath.org/authors/?q=ai:kremer.dustin"Scheffler, Hans-Peter"https://zbmath.org/authors/?q=ai:scheffler.hans-peter.1|scheffler.hans-peterSummary: In this paper we construct vector-valued multi operator-stable random measures that behave locally like operator-stable random measures. The space of integrable functions is characterized in terms of a certain quasi-norm. Moreover, a multi operator-stable moving-average representation of a random field is presented which behaves locally like an operator-stable random field which is also operator-self-similar.Stochastic dominance efficient sets and stochastic spanninghttps://zbmath.org/1472.600332021-11-25T18:46:10.358925Z"Arvanitis, Stelios"https://zbmath.org/authors/?q=ai:arvanitis.steliosSummary: We derive sufficient conditions for non-emptiness of the efficient sets for stochastic dominance relations, usually employed in economics and finance. We do so via the concept of stochastic spanning and its characterization by a saddle-type property. Under the appropriate framework, sufficiency takes the form of semicontinuity of a related functional. In some cases, this boils down to weak continuity of the parameterization of the underlying set of probability distributions.New insights on concentration inequalities for self-normalized martingaleshttps://zbmath.org/1472.600342021-11-25T18:46:10.358925Z"Bercu, Bernard"https://zbmath.org/authors/?q=ai:bercu.bernard"Touati, Taieb"https://zbmath.org/authors/?q=ai:touati.taiebSummary: We propose new concentration inequalities for self-normalized martingales. The main idea is to introduce a suitable weighted sum of the predictable quadratic variation and the total quadratic variation of the martingale. It offers much more flexibility and allows us to improve previous concentration inequalities. Statistical applications on autoregressive process, internal diffusion-limited aggregation process, and online statistical learning are also provided.On decoupling in Banach spaceshttps://zbmath.org/1472.600352021-11-25T18:46:10.358925Z"Cox, Sonja"https://zbmath.org/authors/?q=ai:cox.sonja-gisela"Geiss, Stefan"https://zbmath.org/authors/?q=ai:geiss.stefanSummary: We consider decoupling inequalities for random variables taking values in a Banach space \(X\). We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a Haar-type expansion in which only the pre-specified conditional distributions appear. Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not affect the decoupling properties (in particular, it does not affect the constants involved). As a special case, we deal with one-sided moment inequalities for decoupled dyadic (i.e., Paley-Walsh) martingales and show that Burkholder-Davis-Gundy-type inequalities for stochastic integrals of \(X\)-valued processes can be obtained from decoupling inequalities for \(X\)-valued dyadic martingales.Concentration inequalities for bounded functionals via log-Sobolev-type inequalitieshttps://zbmath.org/1472.600362021-11-25T18:46:10.358925Z"Götze, Friedrich"https://zbmath.org/authors/?q=ai:gotze.friedrich-w"Sambale, Holger"https://zbmath.org/authors/?q=ai:sambale.holger"Sinulis, Arthur"https://zbmath.org/authors/?q=ai:sinulis.arthurSummary: In this paper, we prove multilevel concentration inequalities for bounded functionals \(f = f(X_1, \dots , X_n)\) of random variables \(X_1, \dots , X_n\) that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of \(k\)-tensors of higher order differences of \(f\). We provide applications for both dependent and independent random variables. This includes deviation inequalities for empirical processes \(f(X) = \sup_{g \in{\mathcal{F}}} {|g(X)|}\) and suprema of homogeneous chaos in bounded random variables in the Banach space case \(f(X) = \sup_t{\Vert \sum_{i_1 \ne \dots \ne i_d} t_{i_1 \dots i_d} X_{i_1} \cdots X_{i_d}\Vert }_{{\mathcal{B}}} \). The latter application is comparable to earlier results of Boucheron, Bousquet, Lugosi, and Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for \(U\)-statistics with bounded kernels \(h\) and for the number of triangles in an exponential random graph model.The structure of Gaussian minimal bubbleshttps://zbmath.org/1472.600372021-11-25T18:46:10.358925Z"Heilman, Steven"https://zbmath.org/authors/?q=ai:heilman.steven-mSummary: It is shown that \(m\) disjoint sets with fixed Gaussian volumes that partition \(\mathbb{R}^n\) with minimum Gaussian surface area must be \((m-1)\)-dimensional. This follows from a second variation argument using infinitesimal translations. The special case \(m=3\) proves the Double Bubble problem for the Gaussian measure, with an extra technical assumption. That is, when \(m=3\), the three minimal sets are adjacent 120 degree sectors. The technical assumption is that the triple junction points of the minimizing sets have polynomial volume growth. Assuming again the technical assumption, we prove the \(m=4\) Triple Bubble Conjecture for the Gaussian measure. Our methods combine the Colding-Minicozzi theory of Gaussian minimal surfaces with some arguments used in the Hutchings-Morgan-Ritoré-Ros proof of the Euclidean Double Bubble Conjecture.A stochastic convolution integral inequalityhttps://zbmath.org/1472.600382021-11-25T18:46:10.358925Z"Makasu, Cloud"https://zbmath.org/authors/?q=ai:makasu.cloudSome new results on policy limit allocationshttps://zbmath.org/1472.600392021-11-25T18:46:10.358925Z"Manesh, Sirous Fathi"https://zbmath.org/authors/?q=ai:manesh.sirous-fathi"Izadi, Muhyiddin"https://zbmath.org/authors/?q=ai:izadi.muhyiddin"Khaledi, Baha-Eldin"https://zbmath.org/authors/?q=ai:khaledi.baha-eldinSummary: Suppose that a policyholder faces \(n\) risks \(X_1,\dots,X_n\) which are insured under the policy limit with the total limit of \(l\). Usually, the policyholder is asked to protect each \(X_i\) with an arbitrary limit of \(l_i\) such that \(\sum^n_{i=1}l_i=l\). If the risks are independent and identically distributed with log-concave cumulative distribution function, using the notions of majorization and stochastic orderings, we prove that the equal limits provide the maximum of the expected utility of the wealth of policyholder. If the risks with log-concave distribution functions are independent and ordered in the sense of the reversed hazard rate order, we show that the equal limits is the most favourable allocation among the worst allocations. We also prove that if the joint probability density function is arrangement increasing, then the best arranged allocation maximizes the utility expectation of policyholder's wealth. We apply the main results to the case when the risks are distributed according to a log-normal distribution.Exact estimates of the probability of a non-negative unimodal random value hitting special intervals under incomplete informationhttps://zbmath.org/1472.600402021-11-25T18:46:10.358925Z"Stoikova, L. S."https://zbmath.org/authors/?q=ai:stoikova.l-sSummary: Exact lower estimates are found for the probability that non-negative unimodal random variables \(\mu\) get in the intervals (\(m - \alpha \sigma_\mu \), \(m + \alpha \sigma_\mu\)), where the mode \(m\), which coincides with fixed first moment of random variable \(\mu\), is less than the root-mean-square deviation: \(m < \sigma_\mu \). The parameter \(\alpha\) satisfies the inequalities \(0 < \alpha < m / \sigma_\mu < 1\). The results of this study may be useful in evaluating the probability of hitting the projectile area in target shooting.Concentration inequalities for random tensorshttps://zbmath.org/1472.600412021-11-25T18:46:10.358925Z"Vershynin, Roman"https://zbmath.org/authors/?q=ai:vershynin.romanLet \(x_1,x_2,\ldots\) be independent random vectors in \(\mathbb{R}^n\) whose coordinates are independent random variables with zero mean and unit variance, and let \(X=x_1\otimes\cdots\otimes x_d\), a random tensor in \(\mathbb{R}^{n^d}\). The author proves two concentration inequalities for \(X\). Firstly, in the case where the \(x_k\) are bounded almost surely, it is shown that for a convex and Lipschitz function \(f\), and all \(0\leq t\leq2(\mathbb{E}|f(X)|^2)^{1/2}\), we have
\[
\mathbb{P}\left(\big|f(X)-\mathbb{E}f(X)\big|>t\right)\leq2\exp\left(-\frac{ct^2}{dn^{d-1}\|f\|^2_{Lip}}\right)\,,
\]
for some constant \(c>0\) depending on the bound for the \(x_k\). Secondly, in the case where the \(x_k\) are sub-Gaussian, it is shown that for a linear operator \(A\) taking values in a Hilbert space \(H\), and all \(0\leq t\leq2\|A\|_{HS}\), we have
\[
\mathbb{P}\left(\big|\|AX\|_H-\|A\|_{HS}\big|\geq t\right)\leq2\exp\left(-\frac{ct^2}{dn^{d-1}\|A\|^2_{op}}\right)\,,
\]
where \(c>0\) again depends on the \(x_k\), and where \(\|A\|_{HS}\) and \(\|A\|_{op}\) are the Hilbert-Schmidt and operator norms of \(A\), respectively. As an application of this latter concentration bound, the author shows that random tensors are well conditioned; that is, if \(d=o(\sqrt{n/\log(n)})\) then with high probability \((1-o(1))n^d\) independent copies of \(X\) are far from linearly dependent.Convergence in mean and central limit theorems for weighted sums of martingale difference random vectors with infinite \(r\)th momentshttps://zbmath.org/1472.600422021-11-25T18:46:10.358925Z"Dung, L. V."https://zbmath.org/authors/?q=ai:dung.le-viet"Son, T. C."https://zbmath.org/authors/?q=ai:son.tran-cao|son.ta-cong"Tu, T. T."https://zbmath.org/authors/?q=ai:tu.teng-tao|tu.ton-thatSummary: Let \((X_{nj};1\leq j\leq m_n,n\geq 1)\) be an array of rowwise \(\mathbb{R}^d\)-valued martingale difference \((d\geq 1)\) with respect to \(\sigma\)-fields \((\mathcal{F}_{nj};0\leq j\leq m_n,n\geq 1)\) and let \((C_{nj};1\leq j\leq m_n,n\geq 1)\) be an array of \(m\times d\) matrices of real numbers, where \((m_n;n\geq 1)\) is a sequence of positive integers such that \(m_n\rightarrow\infty\) as \(n\rightarrow\infty\). The aim of this paper is to establish convergence in mean and central limit theorems for weighted sums type \(S_n=\sum_{j=1}^{m_n}C_{nj}X_{nj}\) under some conditions of slow variation at infinity. We also apply the obtained results to study the asymptotic properties of estimates in some statistical models. In addition, two illustrative examples and their simulation are given. This study is motivated by models arising in economics, telecommunications, hydrology, and physics applications where the innovations are often dependent on each other and have infinite variances.Stein's method of normal approximation for dynamical systemshttps://zbmath.org/1472.600432021-11-25T18:46:10.358925Z"Hella, Olli"https://zbmath.org/authors/?q=ai:hella.olli"Leppänen, Juho"https://zbmath.org/authors/?q=ai:leppanen.juho"Stenlund, Mikko"https://zbmath.org/authors/?q=ai:stenlund.mikkoLimit theorems for numbers of returns in arrays under \(\phi\)-mixinghttps://zbmath.org/1472.600442021-11-25T18:46:10.358925Z"Kifer, Yuri"https://zbmath.org/authors/?q=ai:kifer.yuriScaling limits and fluctuations for random growth under capacity rescalinghttps://zbmath.org/1472.600452021-11-25T18:46:10.358925Z"Liddle, George"https://zbmath.org/authors/?q=ai:liddle.george"Turner, Amanda"https://zbmath.org/authors/?q=ai:turner.amanda-gSummary: We evaluate a strongly regularised version of the Hastings-Levitov model \(\mathrm{HL}(\alpha)\) for \(0\leq\alpha<2\). Previous results have concentrated on the small-particle limit where the size of the attaching particle approaches zero in the limit. However, we consider the case where we rescale the whole cluster by its capacity before taking limits, whilst keeping the particle size fixed. We first consider the case where \(\alpha=0\) and show that under capacity rescaling, the limiting structure of the cluster is not a disk, unlike in the small-particle limit. Then we consider the case where \(0<\alpha<2\) and show that under the same rescaling the cluster approaches a disk. We also evaluate the fluctuations and show that, when represented as a holomorphic function, they behave like a Gaussian field dependent on \(\alpha\). Furthermore, this field becomes degenerate as \(\alpha\) approaches 0 and 2, suggesting the existence of phase transitions at these values.Central limit theorems from the roots of probability generating functionshttps://zbmath.org/1472.600462021-11-25T18:46:10.358925Z"Michelen, Marcus"https://zbmath.org/authors/?q=ai:michelen.marcus"Sahasrabudhe, Julian"https://zbmath.org/authors/?q=ai:sahasrabudhe.julianSummary: For each \(n\), let \(X_n \in \{0, \ldots, n \}\) be a random variable with mean \(\mu_n\), standard deviation \(\sigma_n\), and let \[P_n(z) = \sum_{k = 0}^n \mathbb{P}(X_n = k) z^k,\] be its probability generating function. We show that if none of the complex zeros of the polynomials \(\{P_n(z) \}\) is contained in a neighborhood of \(1 \in \mathbb{C}\) and \(\sigma_n > n^\varepsilon\) for some \(\varepsilon > 0\), then \(X_n^\ast = (X_n - \mu_n) \sigma_n^{- 1}\) is asymptotically normal as \(n \to \infty\): that is, it tends in distribution to a random variable \(Z \sim \mathcal{N}(0, 1)\). On the other hand, we show that there exist sequences of random variables \(\{X_n \}\) with \(\sigma_n > C \log n\) for which \(P_n(z)\) has no roots near 1 and \(X_n^\ast\) is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of \(P_n(z)\) and the distribution of the random variable \(X_n\).Poisson approximation for the distribution of the frequency of a given pattern in the outcome sequence of the MCV-generatorhttps://zbmath.org/1472.600472021-11-25T18:46:10.358925Z"Mikhaĭlov, V. G."https://zbmath.org/authors/?q=ai:mikhajlov.v-gSummary: Poisson limit theorem for the distribution of the number of occurrences of a given non-overlapping pattern in the output sequence of the MCV-generator is proved along with the estimate of the convergence rate.On weak law of large numbers for sums of negatively superadditive dependent random variableshttps://zbmath.org/1472.600482021-11-25T18:46:10.358925Z"Naderi, Habib"https://zbmath.org/authors/?q=ai:naderi.habib"Matuła, Przemysław"https://zbmath.org/authors/?q=ai:matula.przemyslaw"Salehi, Mahdi"https://zbmath.org/authors/?q=ai:salehi.mahdi"Amini, Mohammad"https://zbmath.org/authors/?q=ai:amini-dehak.mohammadSummary: In this paper, we extend Kolmogorov-Feller weak law of large numbers for maximal weighted sums of negatively superadditive dependent (NSD) random variables. In addition, we make a simulation study for the asymptotic behavior in the sense of convergence in probability for weighted sums of NSD random variables.Total variation estimates in the Breuer-Major theoremhttps://zbmath.org/1472.600492021-11-25T18:46:10.358925Z"Nualart, David"https://zbmath.org/authors/?q=ai:nualart.david"Zhou, Hongjuan"https://zbmath.org/authors/?q=ai:zhou.hongjuanSummary: This paper provides estimates for the convergence rate of the total variation distance in the framework of the Breuer-Major theorem, assuming some smoothness properties of the underlying function. The results are proved by applying new bounds for the total variation distance between a random variable expressed as a divergence and a standard Gaussian random variable, which are derived by a combination of techniques of Malliavin calculus and Stein's method. The representation of a functional of a Gaussian sequence as a divergence is established by introducing a shift operator on the expansion in Hermite polynomials. Some applications to the asymptotic behavior of power variations of the fractional Brownian motions and to the estimation of the Hurst parameter using power variations are presented.Quantum fluctuations and large-deviation principle for microscopic currents of free fermions in disordered mediahttps://zbmath.org/1472.600502021-11-25T18:46:10.358925Z"Bru, Jean-Bernard"https://zbmath.org/authors/?q=ai:bru.jean-bernard"de Siqueira Pedra, Walter"https://zbmath.org/authors/?q=ai:de-siqueira-pedra.walter"Ratsimanetrimanana, Antsa"https://zbmath.org/authors/?q=ai:ratsimanetrimanana.antsaSummary: We extend the large-deviation results obtained by \textit{N. J. B. Aza} and the present authors [J. Math. Pures Appl. (9) 125, 209--246 (2019; Zbl 1419.82058)] on atomic-scale conductivity theory of free lattice fermions in disordered media. Disorder is modeled by a random external potential, as in the celebrated Anderson model, and a nearest-neighbor hopping term with random complex-valued amplitudes. In accordance with experimental observations, via the large-deviation formalism, our previous paper showed in this case that quantum uncertainty of microscopic electric current densities around their (classical) macroscopic value is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. Here, the quantum fluctuations of linear response currents are shown to exist in the thermodynamic limit, and we mathematically prove that they are related to the rate function of the large-deviation principle associated with current densities. We also demonstrate that, in general, they do not vanish (in the thermodynamic limit), and the quantum uncertainty around the macroscopic current density disappears exponentially fast with an exponential rate proportional to the squared deviation of the current from its macroscopic value and the inverse current fluctuation, with respect to growing space (volume) scales.Fractional moments of the stochastic heat equationhttps://zbmath.org/1472.600512021-11-25T18:46:10.358925Z"Das, Sayan"https://zbmath.org/authors/?q=ai:das.sayan-kumar"Tsai, Li-Cheng"https://zbmath.org/authors/?q=ai:tsai.li-chengSummary: Consider the solution \(\mathcal{Z}(t,x)\) of the one-dimensional stochastic heat equation, with a multiplicative spacetime white noise, and with the delta initial data \(Z(0,x)=\delta(x)\). For any real \(p>0\), we obtained detailed estimates of the \(p\)th moment of \(e^{t/12}\mathcal{Z}(2t,0)\), as \(t\to\infty\), and from these estimates establish the one-point upper-tail large deviation principle of the Kardar-Parisi-Zhang equation. The deviations have speed \(t\) and rate function \(\Phi_+(y)=\frac{4}{3}y^{3/2}\). Our result confirms the existing physics predictions [\textit{P. Le Doussal} et al., ``Large deviations for the height in 1D Kardar-Parisi-Zhang growth at late times'', Europhys. Lett. 113, No. 6, Article ID 60004, 6 p. (2016; \url{doi:10.1209/0295-5075/113/60004})] and also [\textit{A. Kamenev} et al., ``Short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation: starting from a parabola'', Phys. Rev. E 94, No. 3, Article ID 032108, 9 p. (2016; \url{doi:10.1103/PhysRevE.94.032108})].Uniform large deviation principles for Banach space valued stochastic evolution equationshttps://zbmath.org/1472.600522021-11-25T18:46:10.358925Z"Salins, Michael"https://zbmath.org/authors/?q=ai:salins.michael"Budhiraja, Amarjit"https://zbmath.org/authors/?q=ai:budhiraja.amarjit-s"Dupuis, Paul"https://zbmath.org/authors/?q=ai:dupuis.paul-gSummary: We prove a large deviation principle (LDP) for a general class of Banach space valued stochastic differential equations (SDEs) that is uniform with respect to initial conditions in bounded subsets of the Banach space. A key step in the proof is showing that a uniform LDP over compact sets is implied by a uniform over compact sets Laplace principle. Because bounded subsets of infinite-dimensional Banach spaces are in general not relatively compact in the norm topology, we embed the Banach space into its double dual and utilize the weak-\( \star\) compactness of closed bounded sets in the double dual space. We prove that a modified version of our SDE satisfies a uniform Laplace principle over weak-\( \star\) compact sets and consequently a uniform over bounded sets LDP. We then transfer this result back to the original equation using a contraction principle. The main motivation for this uniform LDP is to generalize results of Freidlin and Wentzell concerning the behavior of finite-dimensional SDEs. Here we apply the uniform LDP to study the asymptotics of exit times from bounded sets of Banach space valued small noise SDE, including reaction diffusion equations with multiplicative noise and two-dimensional stochastic Navier-Stokes equations with multiplicative noise.Contraction principle for trajectories of random walks and Cramér's theorem for kernel-weighted sumshttps://zbmath.org/1472.600532021-11-25T18:46:10.358925Z"Vysotsky, Vladislav"https://zbmath.org/authors/?q=ai:vysotsky.vladislav-vSummary: In 2013 \textit{A. A. Borovkov} and \textit{A. A. Mogulskii} [Theory Probab. Appl. 57, No. 1, 1--27 (2013; Zbl 1279.60037); translation from Teor. Veroyatn. Primen. 57, No. 1, 3--34 (2012)] proved a weaker-than-standard ``metric'' large deviations principle (LDP) for trajectories of random walks in \(\mathbb{R}^d\) whose increments have the Laplace transform finite in a neighbourhood of zero. We prove that general metric LDPs are preserved under uniformly continuous mappings. This allows us to transform the result of Borovkov and Mogulskii into standard LDPs. We also give an explicit integral representation of the rate function they found. As an application, we extend the classical Cramér theorem by proving an LPD for kernel-weighted sums of i.i.d. random vectors in \(\mathbb{R}^d\).Large deviations in discrete-time renewal theoryhttps://zbmath.org/1472.600542021-11-25T18:46:10.358925Z"Zamparo, Marco"https://zbmath.org/authors/?q=ai:zamparo.marcoSummary: We establish sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. The framework we consider is the pinning model of polymers, which amounts to a Gibbs change of measure of a classical renewal process and includes it as a special case. We first tackle the problem in a constrained pinning model, where one of the renewals occurs at a given time, by an argument based on convexity and super-additivity. We then transfer the results to the original pinning model by resorting to conditioning.Retraction notice to: ``Convergence of weighted sums for arrays of negatively dependent random variables and its applications''https://zbmath.org/1472.600552021-11-25T18:46:10.358925Z"Baek, Jong-Il"https://zbmath.org/authors/?q=ai:baek.jong-il"Park, Sung-Tae"https://zbmath.org/authors/?q=ai:park.sung-taeFrom the text: This article has been retracted at the request of Editors.
The article [Zbl 1253.60034] is a duplicate of a paper that has already been published in [the authors, J. Theor. Probab. 23, No. 2, 362--377 (2010; Zbl 1196.60045)].
One of the conditions of submission of a paper for publication is that authors declare explicitly that the paper is not under consideration for publication elsewhere. As such this article represents a severe abuse of the scientific publishing system. The scientific community takes a very strong view on this matter and apologies are offered to readers of the journal that this was not detected during the submission process.On exact laws of large numbers for Oppenheim expansions with infinite meanhttps://zbmath.org/1472.600562021-11-25T18:46:10.358925Z"Giuliano, Rita"https://zbmath.org/authors/?q=ai:giuliano.rita"Hadjikyriakou, Milto"https://zbmath.org/authors/?q=ai:hadjikyriakou.miltoSummary: In this work, we investigate the asymptotic behaviour of weighted partial sums of a particular class of random variables related to Oppenheim series expansions. More precisely, we verify convergence in probability as well as almost sure convergence to a strictly positive and finite constant without assuming any dependence structure or the existence of means. Results of this kind are known as \textit{exact weak} and \textit{exact strong} laws.Strong laws of large numbers for arrays of random variables and stable random fieldshttps://zbmath.org/1472.600572021-11-25T18:46:10.358925Z"Nane, Erkan"https://zbmath.org/authors/?q=ai:nane.erkan"Xiao, Yimin"https://zbmath.org/authors/?q=ai:xiao.yimin"Zeleke, Aklilu"https://zbmath.org/authors/?q=ai:zeleke.akliluSummary: Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions for SLLN are described in terms of the \(p\)-th moments of the partial sums of the random fields, which are convenient to verify. The main technical tool in this paper is a maximal inequality for the moments of partial sums of random fields that extends the technique of \textit{S. Chobanyan} et al. [Electron. Commun. Probab. 10, 218--222 (2005; Zbl 1112.60024)] for a sequence of random variables indexed by a one-parameter.Random permutations without macroscopic cycleshttps://zbmath.org/1472.600582021-11-25T18:46:10.358925Z"Betz, Volker"https://zbmath.org/authors/?q=ai:betz.volker"Schäfer, Helge"https://zbmath.org/authors/?q=ai:schafer.helge"Zeindler, Dirk"https://zbmath.org/authors/?q=ai:zeindler.dirkSummary: We consider uniform random permutations of length \(n\) conditioned to have no cycle longer than \(n^\beta\) with \(0 < \beta < 1\), in the limit of large \(n\). Since in unconstrained uniform random permutations most of the indices are in cycles of macroscopic length, this is a singular conditioning in the limit. Nevertheless, we obtain a fairly complete picture about the cycle number distribution at various lengths. Depending on the scale at which cycle numbers are studied, our results include Poisson convergence, a central limit theorem, a shape theorem and two different functional central limit theorems.Quenched invariance principle for a class of random conductance models with long-range jumpshttps://zbmath.org/1472.600592021-11-25T18:46:10.358925Z"Biskup, Marek"https://zbmath.org/authors/?q=ai:biskup.marek|biskup.marek-tomasz"Chen, Xin"https://zbmath.org/authors/?q=ai:chen.xin.1"Kumagai, Takashi"https://zbmath.org/authors/?q=ai:kumagai.takashi"Wang, Jian"https://zbmath.org/authors/?q=ai:wang.jian.2Summary: We study random walks on \({\mathbb{Z}}^d\) (with \(d\ge 2\)) among stationary ergodic random conductances \(\{C_{x,y}:x,y\in{\mathbb{Z}}^d\}\) that permit jumps of arbitrary length. Our focus is on the quenched invariance principle (QIP) which we establish by a combination of corrector methods, functional inequalities and heat-kernel technology assuming that the \(p\)-th moment of \(\sum_{x\in{\mathbb{Z}}^d}C_{0,x}|x|^2\) and \(q\)-th moment of \(1/C_{0,x}\) for \(x\) neighboring the origin are finite for some \(p,q\ge 1\) with \(p^{-1}+q^{-1}<2/d\). In particular, a QIP thus holds for random walks on long-range percolation graphs with connectivity exponents larger than \(2d\) in all \(d\ge 2\), provided all the nearest-neighbor edges are present. Although still limited by moment conditions, our method of proof is novel in that it avoids proving everywhere-sublinearity of the corrector. This is relevant because we show that, for long-range percolation with exponents between \(d+2\) and \(2d\), the corrector exists but fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in \(d\ge 3\) under the conditions complementary to those of the recent work of \textit{P. Bella} and \textit{M. Schäffner} [Ann. Probab. 48, No. 1, 296--316 (2020; Zbl 1450.60064)]. These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.Scaling limits of multi-type Markov branching treeshttps://zbmath.org/1472.600602021-11-25T18:46:10.358925Z"Haas, Bénédicte"https://zbmath.org/authors/?q=ai:haas.benedicte"Stephenson, Robin"https://zbmath.org/authors/?q=ai:stephenson.robinSummary: We introduce multi-type Markov Branching trees, which are simple random population tree models where individuals are characterized by their size and their type and give rise to (size,type)-children in a Galton-Watson fashion, with the rule that the size of any individual is at least the sum of the sizes of its children. Assuming that the macroscopic size-splittings are rare, we describe the scaling limits of multi-type Markov Branching trees in terms of multi-type self-similar fragmentation trees. We observe three different regimes according to whether the probability of type change of a size-biased child is proportional to the probability of macroscopic splitting (the critical regime, in which we get in the limit multi-type fragmentation trees with indeed several types), smaller than the probability of macroscopic splitting (the solo regime, in which the limit trees are monotype as we never see a type change), or larger than the probability of macroscopic splitting (the mixing regime, in which case the types mix in the limit and we get monotype fragmentation trees). This framework allows us to unify models which may a priori seem quite different, a strength which we illustrate with two notable applications. The first one concerns the description of the scaling limits of growing models of random trees built by gluing at each step on the current structure a finite tree picked randomly in a finite alphabet of trees, extending Rémy's well-known algorithm for the generation of uniform binary trees to a fairly broad framework. We are then either in the critical regime with multi-type fragmentation trees in the scaling limit, or in the solo regime. The second application concerns the scaling limits of large multi-type critical Galton-Watson trees when the offspring distributions all have finite second moments. This topic has already been studied but our approach gives a different proof and we improve on previous results by relaxing some hypotheses. We are then in the mixing regime: the scaling limits are always multiple of the Brownian CRT, a pure monotype fragmentation tree in our framework.Martingale spaces and representations under absolutely continuous changes of probabilityhttps://zbmath.org/1472.600612021-11-25T18:46:10.358925Z"Aksamit, Anna"https://zbmath.org/authors/?q=ai:aksamit.anna"Fontana, Claudio"https://zbmath.org/authors/?q=ai:fontana.claudioSummary: In a fully general setting, we study the relation between martingale spaces under two locally absolutely continuous probabilities and prove that the martingale representation property (MRP) is always stable under locally absolutely continuous changes of probability. Our approach relies on minimal requirements, is constructive and, as shown by a simple example, enables us to study situations which cannot be covered by the existing theory.Gaussian process optimization with failures: classification and convergence proofhttps://zbmath.org/1472.600622021-11-25T18:46:10.358925Z"Bachoc, François"https://zbmath.org/authors/?q=ai:bachoc.francois"Helbert, Céline"https://zbmath.org/authors/?q=ai:helbert.celine"Picheny, Victor"https://zbmath.org/authors/?q=ai:picheny.victorSummary: We consider the optimization of a computer model where each simulation either fails or returns a valid output performance. We first propose a new joint Gaussian process model for classification of the inputs (computation failure or success) and for regression of the performance function. We provide results that allow for a computationally efficient maximum likelihood estimation of the covariance parameters, with a stochastic approximation of the likelihood gradient. We then extend the classical improvement criterion to our setting of joint classification and regression. We provide an efficient computation procedure for the extended criterion and its gradient. We prove the almost sure convergence of the global optimization algorithm following from this extended criterion. We also study the practical performances of this algorithm, both on simulated data and on a real computer model in the context of automotive fan design.Pickands-Piterbarg constants for self-similar Gaussian processeshttps://zbmath.org/1472.600632021-11-25T18:46:10.358925Z"Dębicki, Krzysztof"https://zbmath.org/authors/?q=ai:debicki.krzysztof"Ę."https://zbmath.org/authors/?q=ai:e."Tabiś, Kamil"https://zbmath.org/authors/?q=ai:tabis.kamilSummary: For a centered self-similar Gaussian process \(\{Y(t):t\in[0,\infty)\}\) and \(R\ge 0\) we analyze the asymptotic behavior of
\[
\mathcal{H}_Y^R(T)=\mathbf{E}\exp\left(\sup\limits_{t\in [0,T]}\left(\sqrt{2}\, Y(t)-(1+R)\sigma_Y^2(t)\right)\right)
\]
as \(T\to\infty\). We prove that \(\mathcal{H}_Y^R=\lim_{T\to\infty} \mathcal{H}_Y^R(T)\in(0,\infty)\) for \(R>0\) and
\[
\mathcal{H}_Y=\lim_{T\to\infty}\frac{\mathcal{H}_Y^0(T)}{T^\gamma}\in(0,\infty)
\]
for suitably chosen \(\gamma>0\). Additionally, we find bounds for \(\mathcal{H}_Y^R, R>0\), and a surprising relation between \(\mathcal{H}_Y\) and the classical Pickands constants.Imaginary multiplicative chaos: moments, regularity and connections to the Ising modelhttps://zbmath.org/1472.600642021-11-25T18:46:10.358925Z"Junnila, Janne"https://zbmath.org/authors/?q=ai:junnila.janne"Saksman, Eero"https://zbmath.org/authors/?q=ai:saksman.eero"Webb, Christian"https://zbmath.org/authors/?q=ai:webb.christianSummary: In this article we study imaginary Gaussian multiplicative chaos -- namely a family of random generalized functions which can formally be written as \(e^{iX(x)}\), where \(X\) is a log-correlated real-valued Gaussian field on \(\mathbb{R}^d\), that is, it has a logarithmic singularity on the diagonal of its covariance. We study basic analytic properties of these random generalized functions, such as what spaces of distributions these objects live in, along with their basic stochastic properties, such as moment and tail estimates.
After this, we discuss connections between imaginary multiplicative chaos and the critical planar Ising model, namely that the scaling limit of the spin field of the critical planar XOR-Ising model can be expressed in terms of the cosine of the Gaussian free field, that is, the real part of an imaginary multiplicative chaos distribution. Moreover, if one adds a magnetic perturbation to the XOR-Ising model, then the scaling limit of the spin field can be expressed in terms of the cosine of the sine-Gordon field, which can also be viewed as the real part of an imaginary multiplicative chaos distribution.
The first sections of the article have been written in the style of a review, and we hope that the text will also serve as an introduction to imaginary chaos for an uninitiated reader.Oscillatory Breuer-Major theorem with application to the random corrector problemhttps://zbmath.org/1472.600652021-11-25T18:46:10.358925Z"Nualart, David"https://zbmath.org/authors/?q=ai:nualart.david"Zheng, Guangqu"https://zbmath.org/authors/?q=ai:zheng.guangquSummary: In this paper, we present an oscillatory version of the celebrated Breuer-Major theorem that is motivated by the random corrector problem. As an application, we are able to prove new results concerning the Gaussian fluctuation of the random corrector. We also provide a variant of this theorem involving homogeneous measures.Extremes of a type of locally stationary Gaussian random fields with applications to Shepp statisticshttps://zbmath.org/1472.600662021-11-25T18:46:10.358925Z"Tan, Zhongquan"https://zbmath.org/authors/?q=ai:tan.zhongquan"Zheng, Shengchao"https://zbmath.org/authors/?q=ai:zheng.shengchaoSummary: Let \(\{Z(\tau,s),(\tau,s)\in [a,b]\times [0,T]\}\) with some positive constants \(a,b,T\) be a centered Gaussian random field with variance function \(\sigma^2(\tau,s)\) satisfying \(\sigma^2(\tau,s)=\sigma^2(\tau)\). We first derive the exact tail asymptotics (as \(u\rightarrow\infty)\) for the probability that the maximum \(M_H(T) = \max_{(\tau , s) \in [a, b] \times [0, T]} [Z(\tau , s) / \sigma (\tau )]\) exceeds a given level \(u\), for any fixed \(0< a< b < \infty\) and \(T > 0\); and we further derive the extreme limit law for \(M_H(T)\). As applications of the main results, we derive the exact tail asymptotics and the extreme limit laws for Shepp statistics with stationary Gaussian process, fractional Brownian motion and Gaussian integrated process as inputs.Conjunction probability of smooth centered Gaussian processeshttps://zbmath.org/1472.600672021-11-25T18:46:10.358925Z"Viet-Hung Pham"https://zbmath.org/authors/?q=ai:viet-hung-pham.Summary: In this paper we provide an upper bound for the conjunction probability of independent Gaussian smooth processes, and then, we prove that this bound is a good approximation with exponentially smaller error. Our result confirms the heuristic approximation by Euler characteristic method of \textit{K. J. Worsley} and \textit{K. J. Friston} [Stat. Probab. Lett. 47, No. 2, 135--140 (2000; Zbl 0979.62077)] and also implies the exact value of generalized Pickands constant in a special case. Some results for conjunction probability of correlated processes are also discussed.Hausdorff, large deviation and Legendre multifractal spectra of Lévy multistable processeshttps://zbmath.org/1472.600682021-11-25T18:46:10.358925Z"Le Guével, R."https://zbmath.org/authors/?q=ai:le-guevel.ronan"Lévy Véhel, J."https://zbmath.org/authors/?q=ai:vehel.j-levy|levy-vehel.jacquesIn the article, the authors compute the Hausdorff multifractal spectrum of two versions of multistable Lévy processes. These processes are extensions of classical Lévy processes motion to the case in which the stability exponent \(\alpha\) evolves in time. The spectrum provides a decomposition of the unit interval \([0, 1]\) into an uncountable disjoint union of sets of the Hausdorff dimension one. The authors also compute the increments-based large deviations multifractal spectrum of the multistable Lévy processes with independent increments. It is shown that this spectrum is concave, and thus, it coincides with the Legendre multifractal spectrum, but it is different from the Hausdorff multifractal spectrum. In this view, the multistable Lévy process with independent increments provides an example in which the strong multifractal formalism does not hold.Weyl multifractional Ornstein-Uhlenbeck processes mixed with a gamma distributionhttps://zbmath.org/1472.600692021-11-25T18:46:10.358925Z"Es-Sebaiy, Khalifa"https://zbmath.org/authors/?q=ai:es-sebaiy.khalifa"Farah, Fatima-Ezzahra"https://zbmath.org/authors/?q=ai:farah.fatima-ezzahra"Hilbert, Astrid"https://zbmath.org/authors/?q=ai:hilbert.astridSummary: The aim of this paper is to study the asymptotic behavior of aggregated Weyl multifractional Ornstein-Uhlenbeck processes mixed with Gamma random variables. This allows us to introduce a new class of processes, Gamma-mixed Weyl multifractional Ornstein-Uhlenbeck processes (GWmOU), and study their elementary properties such as Hausdorff dimension, local self-similarity and short-range dependence. We also prove that these processes approach the multifractional Brownian motion.Tempered fractional Poisson processes and fractional equations with \(Z\)-transformhttps://zbmath.org/1472.600702021-11-25T18:46:10.358925Z"Gupta, Neha"https://zbmath.org/authors/?q=ai:gupta.neha"Kumar, Arun"https://zbmath.org/authors/?q=ai:kumar.arun-m|kumar.arun-n"Leonenko, Nikolai"https://zbmath.org/authors/?q=ai:leonenko.nikolai-nSummary: In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using \(z\)-transform. We work on tempered versions of time-fractional Poisson process and space-fractional Poisson processes. We also introduce Gegenbauer type fractional differential equations and their solutions using \(z\)-transform. Our results generalize and complement the results available on fractional Poisson processes in several directions.Stochastic models based on moment matchinghttps://zbmath.org/1472.600712021-11-25T18:46:10.358925Z"Dewilde, Patrick"https://zbmath.org/authors/?q=ai:dewilde.patrick-mSummary: The paper considers interpolating models for non-linear, non-Gauss stochastic variables and processes, given a well-ordered set of moments of increasing order. The proposed models use a characterization with \textit{independent parameters}, much in the style of the Schur-Levinson parametrization for the linear, Gaussian case (a topic to which Tom Kailath made seminal contributions), but very different from it, given the different kind of structured matrices involved (Hankel-like instead of Toeplitz). The paper starts out with a review of the classical Hamburger-Akhiezer-Jacobi parametrization for one stochastic variable, using a (non-classical) dynamical system theory approach. Next, the paper generalizes these results to the multivariable case, and presents a detailed generalized Jacobi-like (independent) parametrization for two variables. Like in the Schur-Levinson case, such parametrizations succeed in characterizing models that interpolate the moment data (given the complexity of the issue, only the 2D case is treated in this paper, but using a method that generalizes to more variables).Ergodicity and accuracy of optimal particle filters for Bayesian data assimilationhttps://zbmath.org/1472.600722021-11-25T18:46:10.358925Z"Kelly, David"https://zbmath.org/authors/?q=ai:kelly.d-t-b"Stuart, Andrew M."https://zbmath.org/authors/?q=ai:stuart.andrew-mSummary: Data assimilation refers to the methodology of combining dynamical models and observed data with the objective of improving state estimation. Most data assimilation algorithms are viewed as approximations of the Bayesian posterior (filtering distribution) on the signal given the observations. Some of these approximations are controlled, such as particle filters which may be refined to produce the true filtering distribution in the large particle number limit, and some are uncontrolled, such as ensemble Kalman filter methods which do not recover the true filtering distribution in the large ensemble limit. Other data assimilation algorithms, such as cycled 3DVAR methods, may be thought of as controlled estimators of the state, in the small observational noise scenario, but are also uncontrolled in general in relation to the true filtering distribution. For particle filters and ensemble Kalman filters it is of practical importance to understand how and why data assimilation methods can be effective when used with a fixed small number of particles, since for many large-scale applications it is not practical to deploy algorithms close to the large particle limit asymptotic. In this paper, the authors address this question for particle filters and, in particular, study their accuracy (in the small noise limit) and ergodicity (for noisy signal and observation) without appealing to the large particle number limit. The authors first overview the accuracy and minorization properties for the true filtering distribution, working in the setting of conditional Gaussianity for the dynamics-observation model. They then show that these properties are inherited by optimal particle filters for any fixed number of particles, and use the minorization to establish ergodicity of the filters. For completeness we also prove large particle number consistency results for the optimal particle filters, by writing the update equations for the underlying distributions as recursions. In addition to looking at the optimal particle filter with standard resampling, they derive all the above results for (what they term) the Gaussianized optimal particle filter and show that the theoretical properties are favorable for this method, when compared to the standard optimal particle filter.Global \(C^1\) regularity of the value function in optimal stopping problemshttps://zbmath.org/1472.600732021-11-25T18:46:10.358925Z"de Angelis, Tiziano"https://zbmath.org/authors/?q=ai:de-angelis.tiziano"Peskir, Goran"https://zbmath.org/authors/?q=ai:peskir.goranSummary: We show that if either the process is strong Feller and the boundary point is probabilistically regular for the stopping set, or the process is strong Markov and the boundary point is probabilistically regular for the interior of the stopping set, then the boundary point is Green regular for the stopping set. Combining this implication with the existence of a continuously differentiable flow of the process we show that the value function is continuously differentiable at the optimal stopping boundary whenever the gain function is so. The derived fact holds both in the parabolic and elliptic case of the boundary value problem under the sole hypothesis of probabilistic regularity of the optimal stopping boundary, thus improving upon known analytic results in the PDE literature, and establishing the fact for the first time in the case of integro-differential equations. The method of proof is purely probabilistic and conceptually simple. Examples of application include the first known probabilistic proof of the fact that the time derivative of the value function in the American put problem is continuous across the optimal stopping boundary.Corrigendum to: ``Stopped processes and Doob's optional sampling theorem''https://zbmath.org/1472.600742021-11-25T18:46:10.358925Z"Grobler, Jacobus J."https://zbmath.org/authors/?q=ai:grobler.jacobus-jTwo amendments to the proof of Theorem 5.1(2) in the author's paper [ibid. 497, No. 1, Article ID 124875, 14 p. (2021; Zbl 1464.60037)] are given.Distribution of martingales with bounded square functionshttps://zbmath.org/1472.600752021-11-25T18:46:10.358925Z"Stolyarov, Dmitriy M."https://zbmath.org/authors/?q=ai:stolyarov.dmitry-m"Vasyunin, Vasily"https://zbmath.org/authors/?q=ai:vasyunin.vasily-i"Zatitskiy, Pavel"https://zbmath.org/authors/?q=ai:zatitskii.pavel-b"Zlotnikov, Ilya"https://zbmath.org/authors/?q=ai:zlotnikov.ilya-kSummary: We study the terminate distribution of a martingale whose square function is bounded. We obtain sharp estimates for the exponential and \(p\)-moments, as well as for the distribution function itself. The proofs are based on the elaboration of the Burkholder method and on the investigation of certain locally concave functions.Single jump filtrations and local martingaleshttps://zbmath.org/1472.600762021-11-25T18:46:10.358925Z"Gushchin, Alexander A."https://zbmath.org/authors/?q=ai:gushchin.alexander-aSummary: A single jump filtration \((\mathcal{F}_t)_{t\in\mathbb{R}_+}\) generated by a random variable \(\gamma\) with values in \(\overline{\mathbb{R}}_+\) on a probability space \((\Omega,\mathcal{F},\mathsf{P})\) is defined as follows: a set \(A\in\mathcal{F}\) belongs to \(\mathcal{F}_t\) if \(A\cap \{\gamma >t\}\) is either \(\varnothing\) or \(\{\gamma >t\} \). A process \(M\) is proved to be a local martingale with respect to this filtration if and only if it has a representation \(M_t=F(t)\1_{\{t<\gamma \}}+L\1_{\{t\geqslant \gamma \}} \), where \(F\) is a deterministic function and \(L\) is a random variable such that \(\mathsf{E}|M_t|<\infty\) and \(\mathsf{E}(M_t)=\mathsf{E}(M_0)\) for every \(t\in \{t\in\mathbb{R}_+:\mathsf{P}(\gamma \geqslant t)>0\} \). This result seems to be new even in a special case that has been studied in the literature, namely, where \(\mathcal{F}\) is the smallest \(\sigma \)-field with respect to which \(\gamma\) is measurable (and then the filtration is the smallest one with respect to which \(\gamma\) is a stopping time). As a consequence, a full description of all local martingales is given and they are classified according to their global behaviour.Equidistribution of random walks on compact groups. II: The Wasserstein metrichttps://zbmath.org/1472.600772021-11-25T18:46:10.358925Z"Borda, Bence"https://zbmath.org/authors/?q=ai:borda.benceSummary: We consider a random walk \(S_k\) with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum \(\sum_{k=1}^Nf(S_k)\) with Hölder continuous test functions \(f\), including the central limit theorem, the law of the iterated logarithm and an almost sure approximation by a Wiener process, provided that the distribution of \(S_k\) converges to the Haar measure in the \(p\)-Wasserstein metric fast enough. As an example, we construct discrete random walks on an irrational lattice on the torus \(\mathbb{R}^d/\mathbb{Z}^d\), and find their precise rate of convergence to uniformity in the \(p\)-Wasserstein metric. The proof uses a new Berry-Esseen type inequality for the \(p\)-Wasserstein metric on the torus, and the simultaneous Diophantine approximation properties of the lattice. These results complement the first part of this paper on random walks with an absolutely continuous component and quantitative ergodic theorems for Borel measurable test functions.
For Part I, see [the author, Ann. Inst. Henri Poincaré, Probab. Stat. 57, No. 1, 54--72 (2021; Zbl 1468.60054)].Limit theorems for a random walk with memory perturbed by a dynamical systemhttps://zbmath.org/1472.600782021-11-25T18:46:10.358925Z"Coletti, Cristian F."https://zbmath.org/authors/?q=ai:coletti.cristian-f"de Lima, Lucas R."https://zbmath.org/authors/?q=ai:de-lima.lucas-r"Gava, Renato J."https://zbmath.org/authors/?q=ai:gava.renato-jacob"Luiz, Denis A."https://zbmath.org/authors/?q=ai:luiz.denis-aSummary: We introduce a new random walk with unbounded memory obtained as a mixture of the elephant random walk and the dynamic random walk, which we call the Dynamic Elephant Random Walk (DERW). As a consequence of this mixture, the distribution of the increments of the resulting random process is time dependent. We prove a strong law of large numbers for the DERW and, in a particular case, we provide an explicit expression for its speed. Finally, we give sufficient conditions for the central limit theorem and the law of the iterated logarithm to hold.
{\copyright 2020 American Institute of Physics}Global observables for RW: law of large numbershttps://zbmath.org/1472.600792021-11-25T18:46:10.358925Z"Dolgopyat, Dmitry"https://zbmath.org/authors/?q=ai:dolgopyat.dmitry"Lenci, Marco"https://zbmath.org/authors/?q=ai:lenci.marco"Nándori, Péter"https://zbmath.org/authors/?q=ai:nandori.peterSummary: We consider the sums \(T_N=\sum_{n=1}^NF(S_n)\) where \(S_n\) is a random walk on \(\mathbb{Z}^d\) and \(F:\mathbb{Z}^d\to\mathbb{R}\) is a global observable, that is, a bounded function which admits an average value when averaged over large cubes. We show that \(T_N\) always satisfies the weak Law of Large Numbers but the strong law fails in general except for one dimensional walks with drift. Under additional regularity assumptions on \(F\), we obtain the Strong Law of Large Numbers and estimate the rate of convergence. The growth exponents which we obtain turn out to be optimal in two special cases: for quasiperiodic observables and for random walks in random scenery.A multi-parameter family of self-avoiding walks on the Sierpiński gaskethttps://zbmath.org/1472.600802021-11-25T18:46:10.358925Z"Otsuka, Takafumi"https://zbmath.org/authors/?q=ai:otsuka.takafumiSummary: In this paper, we construct a multi-parameter family of self-avoiding walks on the Sierpiński gasket. It includes the branching model, the loop-erased random walk and the loop-erased self-repelling walk. We reproduce in a unified manner the proof of the existence of the continuum limit and the self-avoiding property of the limit processes. Our limit processes include not only all the processes obtained from the previously studied self-avoiding walk models, but also the ones that have not been constructed before. While the paths of limit processes appearing in the previous works were self-avoiding or filled the whole space, our family includes continuous processes whose path is self-intersecting but does not fill the whole space.On Borwein's conjectures for planar uniform random walkshttps://zbmath.org/1472.600812021-11-25T18:46:10.358925Z"Zhou, Yajun"https://zbmath.org/authors/?q=ai:zhou.yajunSummary: Let \(p_n(x)=\int _0^{\infty }J_0(xt)[J_0(t)]^nxt\,dt\) be Kluyver's probability density for \(n\)-step uniform random walks in the Euclidean plane. Through connection to a similar problem in two-dimensional quantum field theory, we evaluate the third-order derivative \(p_5^{\prime \prime \prime }(0^+)\) in closed form, thereby giving a new proof for a conjecture of J. M. Borwein. By further analogies to Feynman diagrams in quantum field theory, we demonstrate that \(p_n(x),0\leq x\leq 1\) admits a uniformly convergent Maclaurin expansion for all odd integers \(n\geq 5\), thus settling another conjecture of Borwein.The entrance law of the excursion measure of the reflected process for some classes of Lévy processeshttps://zbmath.org/1472.600822021-11-25T18:46:10.358925Z"Chaumont, Loïc"https://zbmath.org/authors/?q=ai:chaumont.loic"Małecki, Jacek"https://zbmath.org/authors/?q=ai:malecki.jacekSummary: We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumpshttps://zbmath.org/1472.600832021-11-25T18:46:10.358925Z"Friesen, Martin"https://zbmath.org/authors/?q=ai:friesen.martin"Jin, Peng"https://zbmath.org/authors/?q=ai:jin.peng"Rüdiger, Barbara"https://zbmath.org/authors/?q=ai:rudiger.barbaraSummary: We study existence and Besov regularity of densities for solutions to stochastic differential equations with Hölder continuous coefficients driven by a \(d\)-dimensional Lévy process \(Z=(Z(t))_{t\geq 0}\), where, for \(t>0\), the density function \(f_t\) of \(Z(t)\) exists and satisfies, for some \((\alpha_i)_{i=1,\dots,d}\subset (0,2)\) and \(C>0\),
\[
\limsup\limits_{t\to 0}t^{1/\alpha_i}\int_{\mathbb{R}^d}|f_t(z+e_ih)-f_t(z)|dz\leq C|h|,\quad h\in\mathbb{R},i=1,\dots,d.
\]
Here \(e_1,\dots,e_d\) denote the canonical basis vectors in \(\mathbb{R}^d\). The latter condition covers anisotropic \((\alpha_1,\dots,\alpha_d)\)-stable laws but also particular cases of subordinate Brownian motion. To prove our result we use some ideas taken from [\textit{A. Debussche} and \textit{N. Fournier}, J. Funct. Anal. 264, No. 8, 1757--1778 (2013; Zbl 1272.60032)].Characterizations of heat kernel estimates for symmetric non-local Dirichlet forms via resistance formshttps://zbmath.org/1472.600842021-11-25T18:46:10.358925Z"Chen, Sheng-Hui"https://zbmath.org/authors/?q=ai:chen.shenghui"Wang, Jian"https://zbmath.org/authors/?q=ai:wang.jian.2|wang.jian.7|wang.jian.9|wang.jian.3|wang.jian.1|wang.jian.4|wang.jian.5Summary: Motivated by \textit{M. T. Barlow} et al. [Commun. Pure Appl. Math. 58, No. 12, 1642--1677 (2005; Zbl 1083.60060)], we obtain new equivalent conditions for two-sided heat kernel estimates of symmetric non-local Dirichlet forms in terms of resistance forms. Characterizations for upper bounds of heat kernel estimates as well as near diagonal lower bounds of Dirichlet heat kernel estimates are also established. These results can be seen as a complement of the recent studies on heat kernel estimates and parabolic Harnack inequalities for symmetric non-local Dirichlet forms in [\textit{Z.-Q. Chen}, \textit{T. Kumagai} and the second author, Stability of heat kernel estimates for symmetric non-local Dirichlet forms. Providence, RI: American Mathematical Society (AMS) (2021; Zbl 07403473); J. Eur. Math. Soc. (JEMS) 22, No. 11, 3747--3803 (2020; Zbl 1455.35004)].Exact sampling of determinantal point processes without eigendecompositionhttps://zbmath.org/1472.600852021-11-25T18:46:10.358925Z"Launay, Claire"https://zbmath.org/authors/?q=ai:launay.claire"Galerne, Bruno"https://zbmath.org/authors/?q=ai:galerne.bruno"Desolneux, Agnès"https://zbmath.org/authors/?q=ai:desolneux.agnesSummary: Determinantal point processes (DPPs) enable the modeling of repulsion: they provide diverse sets of points. The repulsion is encoded in a kernel \(K\) that can be seen, in a discrete setting, as a matrix storing the similarity between points. The main exact algorithm to sample DPPs uses the spectral decomposition of \(K\), a computation that becomes costly when dealing with a high number of points. Here we present an alternative exact algorithm to sample in discrete spaces that avoids the eigenvalues and the eigenvectors computation. The method used here is innovative, and numerical experiments show competitive results with respect to the initial algorithm.Multiscale functional inequalities in probability: constructive approachhttps://zbmath.org/1472.600862021-11-25T18:46:10.358925Z"Duerinckx, Mitia"https://zbmath.org/authors/?q=ai:duerinckx.mitia"Gloria, Antoine"https://zbmath.org/authors/?q=ai:gloria.antoineSummary: Consider an ergodic stationary random field \(A\) on the ambient space \(\mathbb{R}^d\). In order to establish concentration properties for nonlinear functions \(Z(A)\), it is standard to appeal to functional inequalities like Poincaré or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as \textit{multiscale functional inequalities} and which still imply fine concentration properties, and we develop a constructive approach to such inequalities. We consider random fields that can be viewed as transformations of a product structure, for which the question is reduced to devising approximate chain rules for nonlinear random changes of variables. This approach allows us to cover most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including Gaussian fields with arbitrary covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matérn-type processes, as well as Poisson random tessellations. The obtained multiscale functional inequalities, which we primarily develop here in view of their application to concentration and to quantitative stochastic homogenization, are of independent interest.Clustering indices and decay of correlations in non-Markovian modelshttps://zbmath.org/1472.600872021-11-25T18:46:10.358925Z"Abadi, Miguel"https://zbmath.org/authors/?q=ai:abadi.miguel-natalio"Freitas, Ana Cristina Moreira"https://zbmath.org/authors/?q=ai:freitas.ana-cristina-moreira"Freitas, Jorge Milhazes"https://zbmath.org/authors/?q=ai:freitas.jorge-milhazesMean-field FBSDE and optimal controlhttps://zbmath.org/1472.600882021-11-25T18:46:10.358925Z"Agram, Nacira"https://zbmath.org/authors/?q=ai:agram.nacira"Choutri, Salah Eddine"https://zbmath.org/authors/?q=ai:choutri.salah-eddineSummary: We study optimal control for mean-field forward-backward stochastic differential equations with payoff functionals of mean-field type. Sufficient and necessary optimality conditions in terms of a stochastic maximum principle are derived. As an illustration, we solve an optimal portfolio with mean-field risk minimization problem.Addressing the curse of dimensionality in stochastic dynamics: a Wiener path integral variational formulation with free boundarieshttps://zbmath.org/1472.600892021-11-25T18:46:10.358925Z"Petromichelakis, Ioannis"https://zbmath.org/authors/?q=ai:petromichelakis.ioannis"Kougioumtzoglou, Ioannis A."https://zbmath.org/authors/?q=ai:kougioumtzoglou.ioannis-aSummary: A Wiener path integral variational formulation with free boundaries is developed for determining the stochastic response of high-dimensional nonlinear dynamical systems in a computationally efficient manner. Specifically, a Wiener path integral representation of a marginal or lower-dimensional joint response probability density function is derived. Due to this \textit{a priori} marginalization, the associated computational cost of the technique becomes independent of the degrees of freedom (d.f.) or stochastic dimensions of the system, and thus, the `curse of dimensionality' in stochastic dynamics is circumvented. Two indicative numerical examples are considered for highlighting the capabilities of the technique. The first relates to marine engineering and pertains to a structure exposed to nonlinear flow-induced forces and subjected to non-white stochastic excitation. The second relates to nano-engineering and pertains to a 100-d.f. stochastically excited nonlinear dynamical system modelling the behaviour of large arrays of coupled nano-mechanical oscillators. Comparisons with pertinent Monte Carlo simulation data demonstrate the computational efficiency and accuracy of the developed technique.Berry-Esséen bound for the parameter estimation of fractional Ornstein-Uhlenbeck processeshttps://zbmath.org/1472.600902021-11-25T18:46:10.358925Z"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yong|chen.yong.3|chen.yong.1|chen.yong.4|chen.yong.7|chen.yong.5|chen.yong.2|chen.yong.6"Kuang, Nenghui"https://zbmath.org/authors/?q=ai:kuang.nenghui"Li, Ying"https://zbmath.org/authors/?q=ai:li.ying.2|li.ying.1|li.ying|li.ying.3Nonexponential Sanov and Schilder theorems on Wiener space: BSDEs, Schrödinger problems and controlhttps://zbmath.org/1472.600912021-11-25T18:46:10.358925Z"Backhoff-Veraguas, Julio"https://zbmath.org/authors/?q=ai:veraguas.julio-backhoff|backhoff-veraguas.julio-d"Lacker, Daniel"https://zbmath.org/authors/?q=ai:lacker.daniel"Tangpi, Ludovic"https://zbmath.org/authors/?q=ai:tangpi.ludovicSummary: We derive new limit theorems for Brownian motion, which can be seen as nonexponential analogues of the large deviation theorems of Sanov and Schilder in their Laplace principle forms. As a first application, we obtain novel scaling limits of backward stochastic differential equations and their related partial differential equations. As a second application, we extend prior results on the small-noise limit of the Schrödinger problem as an optimal transport cost, unifying the control-theoretic and probabilistic approaches initiated respectively by \textit{T. Mikami} [Probab. Theory Relat. Fields 129, No. 2, 245--260 (2004; Zbl 1061.58034)] and \textit{C. Léonard} [Discrete Contin. Dyn. Syst. 34, No. 4, 1533--1574 (2014; Zbl 1277.49052)]. Lastly, our results suggest a new scheme for the computation of mean field optimal control problems, distinct from the conventional particle approximation. A key ingredient in our analysis is an extension of the classical variational formula (often attributed to \textit{C. Borell} [Potential Anal. 12, No. 1, 49--71 (2000; Zbl 0976.60065)] or \textit{M. Boué} and \textit{P. Dupuis} [Ann. Probab. 26, No. 4, 1641--1659 (1998; Zbl 0936.60059)]) for the Laplace transform of Wiener measure.Convergence of delay equations driven by a Hölder continuous function of order \(1/3<\beta<1/2\)https://zbmath.org/1472.600922021-11-25T18:46:10.358925Z"Besalú, Mireia"https://zbmath.org/authors/?q=ai:besalu.mireia"Binotto, Giulia"https://zbmath.org/authors/?q=ai:binotto.giulia"Rovira, Carles"https://zbmath.org/authors/?q=ai:rovira.carlesThe purpose of this paper is to consider the differential equation with delay \begin{eqnarray*} x_t^r & = & \eta_0+\int_0^t b(u,x^r_u)du+\int_0^t \sigma (x^r_{u-r}) dy_u, \quad t \in (0,T] \\
x^r_t & = & \eta_0, \quad t \in [-r,0] \end{eqnarray*} The goal of the paper is to prove that it converges almost surely in the supremum norm to the solution of the differential equation without delay \begin{eqnarray*} x_t = \eta_0 + \int_0^t b(u,x_u)du + \int_0^t \sigma (x_u) dy_u, \quad t \in [0,T] \end{eqnarray*} when the delay tends to zero.Particles systems and numerical schemes for mean reflected stochastic differential equationshttps://zbmath.org/1472.600932021-11-25T18:46:10.358925Z"Briand, Philippe"https://zbmath.org/authors/?q=ai:briand.philippe"de Raynal, Paul-Éric Chaudru"https://zbmath.org/authors/?q=ai:chaudru-de-raynal.paul-eric"Guillin, Arnaud"https://zbmath.org/authors/?q=ai:guillin.arnaud"Labart, Céline"https://zbmath.org/authors/?q=ai:labart.celineSummary: This paper is devoted to the study of reflected Stochastic Differential Equations when the constraint is not on the paths of the solution but acts on its law. These reflected equations have been introduced recently in a backward form by \textit{P. Briand} et al. [Ann. Appl. Probab. 28, No. 1, 482--510 (2018; Zbl 1391.60133)] in the context of risk measures. We here focus on the forward version of such reflected equations. Our main objective is to provide an approximation of the solutions with the help of interacting particles systems. This approximation allows to design a numerical scheme for this kind of equations.Stability of stochastic dynamic equations with time-varying delay on time scaleshttps://zbmath.org/1472.600942021-11-25T18:46:10.358925Z"Du, Nguyen Huu"https://zbmath.org/authors/?q=ai:nguyen-huu-du."Tuan, Le Anh"https://zbmath.org/authors/?q=ai:tuan.le-anh"Dieu, Nguyen Thanh"https://zbmath.org/authors/?q=ai:dieu.nguyen-thanhSummary: The aim of this article is to consider the existence, uniqueness and uniformly exponential \(p\)-stability of the solution for \(\nabla\)-delay stochastic dynamic equations on time scales via Lyapunov functions. This work can be considered as a unification and generalization of stochastic difference and stochastic differential time-varying delay equations.Distribution tails for solutions of SDE driven by an asymmetric stable Lévy processhttps://zbmath.org/1472.600952021-11-25T18:46:10.358925Z"Eon, Richard"https://zbmath.org/authors/?q=ai:eon.richard"Gradinaru, Mihai"https://zbmath.org/authors/?q=ai:gradinaru.mihaiSummary: The behaviour of the tails of the invariant distribution for stochastic differential equations driven by an asymmetric stable Lévy process is obtained. We generalize a result by \textit{G. Samorodnitsky} and \textit{M. Grigoriu} [Stochastic Processes Appl. 105, No. 1, 69--97 (2003; Zbl 1075.60540)] where the stable driving noise was supposed to be symmetric.Weak Poincaré inequalities for convergence rate of degenerate diffusion processeshttps://zbmath.org/1472.600962021-11-25T18:46:10.358925Z"Grothaus, Martin"https://zbmath.org/authors/?q=ai:grothaus.martin"Wang, Feng-Yu"https://zbmath.org/authors/?q=ai:wang.fengyu|wang.feng-yuThe authors introduce weak Poincaré inequalities for the symmetric and antisymmetric part of the generator to estimate the convergence rate for general degenerate diffusion semigroups. They also present a general result on the weak hypocoercivity for \(C_0\)-semigroups on Hilbert spaces. The main result of the paper applies to a large class of degenerate SDEs, and the state space of the Markov process associated to the semigroup can be very general. In particular, the result applies to degenerate spherical velocity Langevin equations.
The results are important from the viewpoint of applications because solutions to SDEs studied in this paper arise, for instance, in industrial mathematics as so-called fiber laydown processes. They are used as surrogate models for the production process of nonwovens. Here, the rate of convergence to equilibrium is related to the quality of the nonwovens, and so is of practical interest. So, cases in which empirical measurements indicate slow growing potentials are also subsumed as special cases of the model considered in this paper.Edgeworth expansion for Euler approximation of continuous diffusion processeshttps://zbmath.org/1472.600972021-11-25T18:46:10.358925Z"Podolskij, Mark"https://zbmath.org/authors/?q=ai:podolskij.mark"Veliyev, Bezirgen"https://zbmath.org/authors/?q=ai:veliyev.bezirgen"Yoshida, Nakahiro"https://zbmath.org/authors/?q=ai:yoshida.nakahiroSummary: In this paper we present the Edgeworth expansion for the Euler approximation scheme of a continuous diffusion process driven by a Brownian motion. Our methodology is based upon a recent work [\textit{N. Yoshida}, Stochastic Processes Appl. 123, No. 3, 887--933 (2013; Zbl 1261.60034)], which establishes Edgeworth expansions associated with asymptotic mixed normality using elements of Malliavin calculus. Potential applications of our theoretical results include higher order expansions for weak and strong approximation errors associated to the Euler scheme, and for studentized version of the error process.On initial value and terminal value problems for subdiffusive stochastic Rayleigh-Stokes equationhttps://zbmath.org/1472.600982021-11-25T18:46:10.358925Z"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Ngoc, Tran Bao"https://zbmath.org/authors/?q=ai:ngoc.tran-bao"Thach, Tran Ngoc"https://zbmath.org/authors/?q=ai:thach.tran-ngoc"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan.The authors study the initial value problem and the terminal value problem for a stochastic time-fractional Rayleigh-Stokes equation, where the source function and the time-spatial noise are nonlinear. The stochastic part is introduced by Wiener process and the fractional derivatives are taken in the sense of Riemann-Liouville. The source function and the time-spatial noise satisfy the globally Lipschitz conditions. \newline The authors provide some existence results and regularity properties for the mild solution of each problem.Weak convergence and invariant measure of a full discretization for parabolic SPDEs with non-globally Lipschitz coefficientshttps://zbmath.org/1472.600992021-11-25T18:46:10.358925Z"Cui, Jianbo"https://zbmath.org/authors/?q=ai:cui.jianbo"Hong, Jialin"https://zbmath.org/authors/?q=ai:hong.jialin"Sun, Liying"https://zbmath.org/authors/?q=ai:sun.liyingIn this paper, the authors propose a full discretization to
approximate the invariant measure numerically for parabolic
SPDEs with non-globally Lipschitz coefficients of the type:
\[
d X(t) = ( A X(t) + F( X(t)) ) dt + d W(t), \quad t > 0,
\qquad \text{with} \quad X(0) = X_0,
\]
where $A : Dom(A) \subset {\mathbb H} := L^2({\mathcal O}) \to {\mathbb H}$ is the Laplacian operator on ${\mathcal O} = [0, L]^d$, $d \leqslant 3$, $L > 0$, under homogeneous Dirichlet
boundary condition, $F$ is the Nemytskii operator of a real-valued
one-sided Lipschitz function $f$, i.e., $F(X)(\xi) = f( X(\xi))$, and
$W(t)$, $t \geq 0$ is a generalized $Q$-Wiener process on a
filtered probability space $( \Omega, {\mathcal F}, {\mathbb P}, \{ {\mathcal F}_t \}_{ t \geq 0} )$.
Note that $Q$ is a bounded, linear, self-adjoint and positive definite
operator on ${\mathbb H}$ and satisfies
\[
\left\Vert ( - A)^{ \frac{ \beta -1}{2} } \right\Vert_{ {\mathcal L}_2^0 } < \infty \qquad \text{with} \quad 0 < \beta \leqslant 2
\]
with ${\mathcal L}_2^0 := {\mathcal L}_2 ( U_0, {\mathbb H})$ (= the space of Hilbert-Schmidt operators),
$U_0 = Q^{1/2} ( {\mathbb H} )$.
One of the peculiar features in their work consists in derivation
of a priori estimates and regularity estimates of the numerical
solution via a variational approach and Malliavin calculus.
Under certain hypotheses, they present the time-independent regularity estimates for the corresponding Kolmogorov
equation and the time-independent weak convergence analysis
for the full discretization:
\[
X_{k+1}^N = S_{\delta t} X_k^N + \delta t S_{\delta t} P^N F( X_{k+1}^N ) + S_{\delta t} P^N \delta W_k,
\]
where $S_{\delta t} = ( I - A \delta t)^{-1}$, $N$ is the dimension of the spectral Galerkin projection space and $\delta t$ is the time-step size, $X_0^N = P^N X_0$ and
$P^N \delta W_k = P^N ( W( (k+1) \delta t ) - W( k \delta t))$.
The following is a result on weak convergence rate for (2) (see the paper):
Theorem 1. Assume that $X_0$ is a sufficiently smooth
function. Let $f$ be a cubic polynomial. Assume that $T > 0$ and
\[
\delta t_0 \in ( 0, 1 \wedge 1/ \{ ( 2 \lambda_F - 2 \lambda_1) \vee 0 \} ).
\]
Then for any $\phi \in C_b^2({\mathbb H})$, there exists
$C( T, X_0, Q, \phi) > 0$ such that for any $\delta t \in (0,
\delta t_0 ]$, $K \delta t = T$, $K \in {\mathbb N}^T$ and $N \in {\mathbb N}^T$,
\[
\vert \, {\mathbb E} [ \phi( X(T)) - \phi( X_K^N) ] \, \vert \leqslant C(T, X_0, Q, \phi) ( \delta t^{\gamma} + \lambda_N^{- \gamma} ),
\]
where $\lambda_1$ is the smallest eigenvalue of $-A$, and
$\lambda_F$ is the one-sided Lipschitz constant of $F$.
Furthermore, it is shown that the $V$-uniformly ergodic invariant measure of the original system is approximated by this full
discretization with weak convergence rate.
The following is about the time-independent weak error estimate:
for any $\phi \in C_b^2({\mathbb H})$, there exists
$C(X_0, Q, \phi) > 0$ such that for $\delta t \in (0, \delta t_0]$,
$K \geq 2$ and $N \in {\mathbb N}^+$,
\[
\vert \, {\mathbb E} [ \phi( X( K \delta t)) - \phi( X_K^N)]
\, \vert \leqslant C(X_0, Q, \phi) \{ 1 + ( K \delta t)^{- \gamma} \} ( \delta t^{\gamma} + \lambda_N^{- \gamma} )
\]
holds. Lastly, several examples are provided as well that numerical
experiments verify theoretical findings.
\par
For other related works, see, e.g., [\textit{R. Anton} et al., IMA J. Numer. Anal. 40, No. 1, 247--284 (2020; Zbl 1470.80011)] for a fully discrete approximation of stochastic
heat equation, and [\textit{D. Conus} et al., Ann. Appl. Probab. 29, No. 2, 653--716 (2019; Zbl 07047435)] for weak
convergence rates of spectral Galerkin approximations for
SPDEs.The Osgood condition for stochastic partial differential equationshttps://zbmath.org/1472.601002021-11-25T18:46:10.358925Z"Foondun, Mohammud"https://zbmath.org/authors/?q=ai:foondun.mohammud"Nualart, Eulalia"https://zbmath.org/authors/?q=ai:nualart.eulaliaSummary: We study the following equation
\[
\frac{\partial u(t,x)}{\partial t}=\Delta u(t,x)+b\bigl(u(t,x)\bigr)+\sigma \dot{W}(t,x),\quad t>0,
\]
where \(\sigma\) is a positive constant and \(\dot{W}\) is a space-time white noise. The initial condition \(u(0,x)=u_0(x)\) is assumed to be a nonnegative and continuous function. We first study the problem on \([0,1]\) with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of \textit{J. F. Bonder} and \textit{P. Groisman} [Physica D 238, No. 2, 209--215 (2009; Zbl 1173.35543)], our first result shows that the solution blows up in finite time if and only if for some \(a>0\),
\[
\int_a^{\infty }\frac{1}{b(s)}\,\mathrm{d}s<\infty,
\]
which is the well-known Osgood condition. We also consider the same equation on the whole line and show that the above condition is sufficient for the nonexistence of global solutions. Various other extensions are provided; we look at equations with fractional Laplacian and spatial colored noise in \(\mathbf{R}^d \).Infinite delay fractional stochastic integro-differential equations with Poisson jumps of neutral typehttps://zbmath.org/1472.601012021-11-25T18:46:10.358925Z"Hussain, R. Jahir"https://zbmath.org/authors/?q=ai:hussain.r-jahir"Hussain, S. Satham"https://zbmath.org/authors/?q=ai:hussain.s-sathamThe well-posedness and continuous dependence are presented for the mild solution to a class of stochastic neutral integral-differential equations with infinite delay driven by Poisson jumps on a separable Hilbert space \(\mathbb H\):
\[
\begin{aligned}
d P(t,x_t)= &\,\int_0^t \frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)} A P(s, x_s) d s d t\\
&+ f(t,x_t)d t+ \sigma(t,x_t) d W(t)\\
&+ \int_{\mathbb Z} h(t,x_t,y) \tilde N(d t, dy),\\
&\ x_0\in \mathcal B:=C((-\infty,0];\mathbb H), t\in [0,T],
\end{aligned}
\]
where \(T>0\) is a fixed constant, \((A,\mathcal{A})\) is a densely defined linear operator on \(\mathbb H\) of sectorial type, \(W(t)\) is a Wiener process on \(\mathbb H\) with finite trace nuclear covariance, \(\tilde N\) is a compensated Poisson martingale measure over a reference space \(\mathbb Z\), \(x_t\in \mathcal B\) with \(x_t(\theta):= x(t+\theta)\) for \(\theta\in (-\infty,0]\) is the segment of \(x(\cdot)\) up to time \(t\), and \(P, f, \sigma, h\) are proper defined functionals. The main results are illustrated with specific examples.Paracontrolled quasi-geostrophic equation with space-time white noisehttps://zbmath.org/1472.601022021-11-25T18:46:10.358925Z"Inahama, Yuzuru"https://zbmath.org/authors/?q=ai:inahama.yuzuru"Sawano, Yoshihiro"https://zbmath.org/authors/?q=ai:sawano.yoshihiroThe authors consider the stochastic dissipative quasi-geostrophic equations
\[
\partial_tu^\varepsilon=-(-\Delta)^{\theta/2}u^\varepsilon+R\perp u^\varepsilon\cdot\nabla u^\varepsilon+\xi^\varepsilon,\qquad u^\varepsilon(0)=u_0
\]
on the two-dimensional torus \(\mathbb T^2\), where \(\varepsilon\in(0,1)\), \(\theta\in(7/4,2]\), \(R=(R_1,R_2)\) with \(R_j\) being the \(j\)-th Riesz transform on \(\mathbb T^2\), \(\xi\) is an additive space-time white noise on \(\mathbb T^2\) and \(\xi^\varepsilon\) for \(\varepsilon>0\) are mollifications of \(\xi \). If \(\kappa>\kappa_\theta\) for an explicitly defined threshold \(\kappa_\theta\) and \(u_0\in\mathcal C^\kappa\) where \(\mathcal C^\kappa\) is the Besov-Hölder space \(B^\kappa_{\infty,\infty}\) on \(\mathbb T^2\) then the unique solutions \(u^\varepsilon\) exist on \([0,T^\varepsilon_*]\) where \(T^\varepsilon_*\) are positive random times, \(T^\varepsilon_*\) converge in probability to a positive random time \(T_*\) as \(\varepsilon\to 0+\), and there exists a \(C^\kappa\)-valued process \(u\) on \([0,T_*)\) such that
\[
\lim_{\varepsilon\to 0+}\left(\sup_{s\in[0,T^\varepsilon_*\land\frac{T_*}{2}]}\|u^\varepsilon_s-u_s\|_{\mathcal C^{\kappa}}\right)=0
\]
in probability. Moreover, \(u\) is independent of the mollification.Large time asymptotic properties of the stochastic heat equationhttps://zbmath.org/1472.601032021-11-25T18:46:10.358925Z"Kohatsu-Higa, Arturo"https://zbmath.org/authors/?q=ai:kohatsu-higa.arturo"Nualart, David"https://zbmath.org/authors/?q=ai:nualart.davidIn this paper, the authors studied the large time asymptotic behavior of the stochastic heat equation. In particular, the authors discussed an extension of these results in the case of space averages on an interval \([-R, R]\) and both \(R\) and \(t\) tend to infinity. Further, the case of nonlinear equations is also discussed.Locally robust random attractors in stochastic non-autonomous magneto-hydrodynamicshttps://zbmath.org/1472.601042021-11-25T18:46:10.358925Z"Li, Fuzhi"https://zbmath.org/authors/?q=ai:li.fuzhi"Yangrong, Li"https://zbmath.org/authors/?q=ai:yangrong.liThis paper shows the local robustness result of random attractors (towards a deterministic attractor), which generalizes some related results in the literature. The main results are illustrated in the content of stochastic non-autonomous magneto-hydrodynamics (MHD) equations. By using joint convergence of the cocycles, collective local compactness and deterministic recurrence of the random attractors, the authors prove that the family of pullback random attractors is locally uniform convergent to the pullback attractor of the deterministic MHD equation when the density of random noise tends to 0.Representation of the solution of Goursat problem for second order linear stochastic hyperbolic differential equationshttps://zbmath.org/1472.601052021-11-25T18:46:10.358925Z"Mansimov, Kamil' Baĭramali"https://zbmath.org/authors/?q=ai:mansimov.kamil-bairamali-oglu"Mastaliev, Rashad Ogtaĭ"https://zbmath.org/authors/?q=ai:mastaliev.rashad-ogtai-oglySummary: The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noisehttps://zbmath.org/1472.601062021-11-25T18:46:10.358925Z"Wang, Xiaohu"https://zbmath.org/authors/?q=ai:wang.xiaohu"Li, Dingshi"https://zbmath.org/authors/?q=ai:li.dingshi"Shen, Jun"https://zbmath.org/authors/?q=ai:shen.junThe authors investigate the asymptotic behavior of the solutions of the stochastic wave equation driven by an additive white noise on unbounded domains and its Wong-Zakai approximation.
The main idea of Wong-Zakai approximation is to use deterministic differential equations to approximate the solutions of the stochastic differential equations.
In the article, Brownian motion is approximated using the Euler approximation. The authors prove the existence and uniqueness of tempered pullback attractors for stochastic wave equation and its Wong-Zakai approximation. Then, they show that the attractor of the Wong-Zakai approximate equation converges to the one of the stochastic wave equation driven by additive noise as the correlation time of noise approaches zero.A systematic path to non-Markovian dynamics: new response probability density function evolution equations under Gaussian coloured noise excitationhttps://zbmath.org/1472.601072021-11-25T18:46:10.358925Z"Mamis, K. I."https://zbmath.org/authors/?q=ai:mamis.k-i"Athanassoulis, G. A."https://zbmath.org/authors/?q=ai:athanassoulis.gerassimos-a"Kapelonis, Z. G."https://zbmath.org/authors/?q=ai:kapelonis.z-gSummary: Determining evolution equations governing the probability density function (pdf) of non-Markovian responses to random differential equations (RDEs) excited by coloured noise, is an important issue arising in various problems of stochastic dynamics, advanced statistical physics and uncertainty quantification of macroscopic systems. In the present work, such equations are derived for a scalar, nonlinear RDE under additive coloured Gaussian noise excitation, through the stochastic Liouville equation. The latter is an exact, yet non-closed equation, involving averages over the time history of the non-Markovian response. This non-locality is treated by applying an extension of the Novikov-Furutsu theorem and a novel approximation, employing a stochastic Volterra-Taylor functional expansion around instantaneous response moments, leading to efficient, closed, approximate equations for the response pdf. These equations retain a tractable amount of non-locality and nonlinearity, and they are valid in both the transient and long-time regimes for any correlation function of the excitation. Also, they include as special cases various existing relevant models, and generalize Hänggi's ansatz in a rational way. Numerical results for a bistable nonlinear RDE confirm the accuracy and the efficiency of the new equations. Extension to the multidimensional case (systems of RDEs) is feasible, yet laborious.On Bernstein processes of maximal entropyhttps://zbmath.org/1472.601082021-11-25T18:46:10.358925Z"Vuillermot, Pierre-A."https://zbmath.org/authors/?q=ai:vuillermot.pierre-aSummary: In this article we define and investigate statistical operators and an entropy functional for Bernstein stochastic processes associated with hierarchies of forward-backward systems of decoupled deterministic linear parabolic partial differential equations. The systems under consideration are defined on open bounded domains \(D\subset\mathbb{R}^d\) of Euclidean space where \(d\in\mathbb{N}^+\) is arbitrary, and are subject to Neumann boundary conditions. We assume that the elliptic part of the parabolic operator in the equations is a self-adjoint Schrödinger operator, bounded from below and with compact resolvent in \(L^2(D)\) The statistical operators we consider are then trace-class operators defined from sequences of probabilities associated with the point spectrum of the elliptic part in question, which allow the distinction between pure and mixed processes. We prove in particular that the Bernstein processes of maximal entropy are those for which the associated sequences of probabilities are of Gibbs type. We illustrate our results by considering processes associated with a specific hierarchy of forward-backward heat equations defined in a two-dimensional disk.Extensions and solutions for nonlinear diffusion equations and random walkshttps://zbmath.org/1472.601092021-11-25T18:46:10.358925Z"Lenzi, E. K."https://zbmath.org/authors/?q=ai:kaminski-lenzi.ervin"Lenzi, M. K."https://zbmath.org/authors/?q=ai:lenzi.marcelo-k"Ribeiro, H. V."https://zbmath.org/authors/?q=ai:ribeiro.haroldo-v"Evangelista, L. R."https://zbmath.org/authors/?q=ai:evangelista.luiz-robertoSummary: We investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by Einstein to explain the Brownian motion. We show here how to properly modify that framework in order to handle different physical scenarios. We obtain solutions for nonlinear diffusion equations that emerge from the random walk approach and analyse possible connections with a generalized thermostatistics formalism. Finally, we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations, depending on the choice of distribution functions related to the spreading of systems.New stochastic operational matrix method for solving stochastic Itô-Volterra integral equations characterized by fractional Brownian motionhttps://zbmath.org/1472.601102021-11-25T18:46:10.358925Z"Saha Ray, S."https://zbmath.org/authors/?q=ai:saha-ray.santanu"Singh, S."https://zbmath.org/authors/?q=ai:singh.soumyendraSummary: In this paper, stochastic integral equations characterized by fractional Brownian motion have been studied. The fractional stochastic integral equation has been solved by second kind Chebyshev wavelets. The convergence and error analysis have been discussed for the efficiency of the discussed method. In addition, two illustrative examples have been solved to examine the efficiency and accuracy of the proposed scheme.Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equationshttps://zbmath.org/1472.601112021-11-25T18:46:10.358925Z"Becker, Sebastian"https://zbmath.org/authors/?q=ai:becker.sebastian"Gess, Benjamin"https://zbmath.org/authors/?q=ai:gess.benjamin"Jentzen, Arnulf"https://zbmath.org/authors/?q=ai:jentzen.arnulf"Kloeden, Peter E."https://zbmath.org/authors/?q=ai:kloeden.peter-erisSummary: This article establishes optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise. Thereby, this work proves the optimality of the strong convergence rates for certain full-discrete approximations of stochastic Allen-Cahn equations with space-time white noise which have been obtained in a recent previous work of the authors of this article.Non-asymptotic Gaussian estimates for the recursive approximation of the invariant distribution of a diffusionhttps://zbmath.org/1472.601122021-11-25T18:46:10.358925Z"Honoré, I."https://zbmath.org/authors/?q=ai:honore.igor"Menozzi, S."https://zbmath.org/authors/?q=ai:menozzi.stephane"Pagès, G."https://zbmath.org/authors/?q=ai:pages.gilles|pages.gaelSummary: We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant distribution \(\nu\) of an ergodic Brownian diffusion process and the empirical distribution of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions \(f\) such that \(f-\nu (f)\) is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when some suitable squared-norms of the diffusion coefficient also belong to this class. We apply these estimates to design computable non-asymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem.A new second-order one-step scheme for solving decoupled FBSDES and optimal error estimateshttps://zbmath.org/1472.601132021-11-25T18:46:10.358925Z"Li, Yang"https://zbmath.org/authors/?q=ai:li.yang.3"Yang, Jie"https://zbmath.org/authors/?q=ai:yang.jie.1|yang.jie.3|yang.jie.4|yang.jie.2"Zhao, Weidong"https://zbmath.org/authors/?q=ai:zhao.weidongSummary: A novel second-order numerical scheme for solving decoupled forward back-ward stochastic differential equations is proposed. Unlike known second-order schemes for such equations, the forward stochastic differential equations are approximated by a simplified weak order-2 Itô-Taylor scheme. This makes the method more implement-able and enhances the accuracy. If the operators involved satisfy certain commutativity conditions, the schemewith quadratic convergence can besimplified, which is important in applications. The stability of the method is studied and second-order optimal error estimates are obtained.An explicit second order scheme for decoupled anticipated forward backward stochastic differential equationshttps://zbmath.org/1472.601142021-11-25T18:46:10.358925Z"Sun, Yabing"https://zbmath.org/authors/?q=ai:sun.yabing"Zhao, Weidong"https://zbmath.org/authors/?q=ai:zhao.weidongSummary: The Feynman-Kac formula and the Lagrange interpolation method are used in the construction of an explicit second order scheme for decoupled anticipated forward backward stochastic differential equations. The stability of the scheme is rigorously proved and error estimates are established. The scheme has the second order accuracy when weak order 2.0 Taylor scheme is employed to solve stochastic differential equations. Numerical tests confirm the theoretical findings.Nonparametric estimation of jump rates for a specific class of piecewise deterministic Markov processeshttps://zbmath.org/1472.601152021-11-25T18:46:10.358925Z"Krell, Nathalie"https://zbmath.org/authors/?q=ai:krell.nathalie"Schmisser, Émeline"https://zbmath.org/authors/?q=ai:schmisser.emelineSummary: In this paper, we consider a unidimensional piecewise deterministic Markov process (PDMP), with homogeneous jump rate \(\lambda (x)\). This process is observed continuously, so the flow \(\phi\) is known. To estimate nonparametrically the jump rate, we first construct an adaptive estimator of the stationary density, then we derive a quotient estimator \(\hat{\lambda}_n\) of \(\lambda \). Under some ergodicity conditions, we bound the risk of these estimators (and give a uniform bound on a small class of functions), and prove that the estimator of the jump rate is nearly minimax (up to a \(\ln^2(n)\) factor). The simulations illustrate our theoretical results.Markov chains with exponential return times are finitaryhttps://zbmath.org/1472.601162021-11-25T18:46:10.358925Z"Angel, Omer"https://zbmath.org/authors/?q=ai:angel.omer"Spinka, Yinon"https://zbmath.org/authors/?q=ai:spinka.yinonSummary: Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an independent and identically distributed (i.i.d.) process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of \(\mathbb{Z}\) is a finitary factor of an i.i.d. process.A threshold for cutoff in two-community random graphshttps://zbmath.org/1472.601172021-11-25T18:46:10.358925Z"Ben-Hamou, Anna"https://zbmath.org/authors/?q=ai:ben-hamou.annaSummary: In this paper, we are interested in the impact of communities on the mixing behavior of the nonbacktracking random walk. We consider sequences of sparse random graphs of size \(N\) generated according to a variant of the classical configuration model which incorporates a two-community structure. The strength of the bottleneck is measured by a parameter \(\alpha\) which roughly corresponds to the fraction of edges that go from one community to the other. We show that if \(\alpha \gg \frac{1}{\log N}\), then the nonbacktracking random walk exhibits cutoff at the same time as in the one-community case, but with a larger cutoff window, and that the distance profile inside this window converges to the Gaussian tail function. On the other hand, if \(\alpha \ll \frac{1}{\log N}\) or \(\alpha \asymp \frac{1}{\log N}\), then the mixing time is of order \(1/\alpha\) and there is no cutoff.Geometric ergodicity of the bouncy particle samplerhttps://zbmath.org/1472.601182021-11-25T18:46:10.358925Z"Durmus, Alain"https://zbmath.org/authors/?q=ai:durmus.alain"Guillin, Arnaud"https://zbmath.org/authors/?q=ai:guillin.arnaud"Monmarché, Pierre"https://zbmath.org/authors/?q=ai:monmarche.pierreSummary: The Bouncy Particle Sampler (BPS) is a Monte Carlo Markov chain algorithm to sample from a target density known up to a multiplicative constant. This method is based on a kinetic piecewise deterministic Markov process for which the target measure is invariant. This paper deals with theoretical properties of BPS. First, we establish geometric ergodicity of the associated semi-group under weaker conditions than in [\textit{G. Deligiannidis} et al., Ann. Stat. 47, No. 3, 1268--1287 (2019; Zbl 1467.60057)] both on the target distribution and the velocity probability distribution. This result is based on a new coupling of the process which gives a quantitative minorization condition and yields more insights on the convergence. In addition, we study on a toy model the dependency of the convergence rates on the dimension of the state space. Finally, we apply our results to the analysis of simulated annealing algorithms based on BPS.Mixing time and eigenvalues of the abelian sandpile Markov chainhttps://zbmath.org/1472.601192021-11-25T18:46:10.358925Z"Jerison, Daniel C."https://zbmath.org/authors/?q=ai:jerison.daniel-c"Levine, Lionel"https://zbmath.org/authors/?q=ai:levine.lionel"Pike, John"https://zbmath.org/authors/?q=ai:pike.johnSummary: The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph \(G\). By viewing this chain as a (nonreversible) random walk on an abelian group, we give a formula for its eigenvalues and eigenvectors in terms of ``multiplicative harmonic functions'' on the vertices of \(G\). We show that the spectral gap of the sandpile chain is within a constant factor of the length of the shortest noninteger vector in the dual Laplacian lattice, while the mixing time is at most a constant times the smoothing parameter of the Laplacian lattice. We find a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on \(G\): If the latter has a sufficiently large spectral gap, then the former has a small gap! In the case where \(G\) is the complete graph on \(n\) vertices, we show that the sandpile chain exhibits cutoff at time \(\frac{1}{4\pi ^2}n^3\log n\).Cutoff for product replacement on finite groupshttps://zbmath.org/1472.601202021-11-25T18:46:10.358925Z"Peres, Yuval"https://zbmath.org/authors/?q=ai:peres.yuval"Tanaka, Ryokichi"https://zbmath.org/authors/?q=ai:tanaka.ryokichi"Zhai, Alex"https://zbmath.org/authors/?q=ai:zhai.alexLet \(G\) be a finite group, \([n]:=\{1,2,\cdots,n\}\), and \(G^n\) be the set of all functions \(\sigma: [n]\to G\). Denote by \(\mathcal{S}\) the space of generating \(n\)-tuples, i.e., the set of \(\sigma\) whose values generate \(G\) as a group:
\[
\mathcal{S}:=\{\sigma\in G^n: \langle \sigma(1),\dots,\sigma(n)\rangle=G\}.
\]
Define the so-called product replacement chain \((\sigma_t)_{t\geq 0}\) on \(\mathcal{S}\) as follows: if we have a current state \(\sigma\), then uniformly at random, choose an ordered pair \((i,j)\) of distinct integers in \([n]\), and change the value of \(\sigma(i)\) to \(\sigma(i)\sigma(j)^{\pm 1}\), where the signs are chosen with equal probability. This paper shows that the total-variation mixing time of the chain has a cutoff at time \(\frac{3}{2}n\log n\) with window of order \(n\) as \(n\to \infty\). This extends a result of \textit{A. Ben-Hamou} and the first author [Electron. Commun. Probab. 23, Paper No. 32, 10 p. (2018; Zbl 1397.60096)] (who proved the result for \(G=\mathbb{Z}/2\)) and confirms a conjecture of \textit{P. Diaconis} and \textit{L. Saloff-Coste} [Invent. Math. 134, No. 2, 251--299 (1998; Zbl 0921.60003)] that for an arbitrary but fixed finite group, the mixing time of the product replacement chain is \(O(n\log n)\).Unified theory for finite Markov chainshttps://zbmath.org/1472.601212021-11-25T18:46:10.358925Z"Rhodes, John"https://zbmath.org/authors/?q=ai:rhodes.john-l"Schilling, Anne"https://zbmath.org/authors/?q=ai:schilling.anneSummary: We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup \(S\). Our methods use geometric finite semigroup theory via the Karnofsky-Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to \(P(n)\), the set of all subsets of an \(n\) element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or \(\mathcal{R}\)-trivial bands) by Brown and Diaconis, extensions to \(\mathcal{R}\)-trivial semigroups by Ayyer, Steinberg, Thiéry and the second author, and important recent work by Chung and Graham. The Karnofsky-Rhodes expansion of the right Cayley graph of \(S\) in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded \(S\). Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded \(S\) and the original \(S\). Analyzing the normal forms also provides an estimate on the mixing time.Normal distributions of finite Markov chainshttps://zbmath.org/1472.601222021-11-25T18:46:10.358925Z"Rhodes, John"https://zbmath.org/authors/?q=ai:rhodes.john-l"Schilling, Anne"https://zbmath.org/authors/?q=ai:schilling.anneCorrection to: ``Carr-Nadtochiy's weak reflection principle for Markov chains on \(\mathbb{Z}^d\)''https://zbmath.org/1472.601232021-11-25T18:46:10.358925Z"Imamura, Yuri"https://zbmath.org/authors/?q=ai:imamura.yuriCorrection to the author's paper [ibid. 38, No. 1, 257--267 (2021; Zbl 1470.60207)].Statistical Taylor series expansion: an approach for epistemic uncertainty propagation in Markov reliability modelshttps://zbmath.org/1472.601242021-11-25T18:46:10.358925Z"Bachi, Katia"https://zbmath.org/authors/?q=ai:bachi.katia"Abbas, Karim"https://zbmath.org/authors/?q=ai:abbas.karim"Heidergott, Bernd"https://zbmath.org/authors/?q=ai:heidergott.bernd-fSummary: In this paper we develop a new Taylor series expansion method for computing model output metrics under epistemic uncertainty in the model input parameters. Specifically, we compute the expected value and the variance of the stationary distribution associated with Markov reliability models. In the multi-parameter case, our approach allows to analyze the impact of correlation between the uncertainty on the individual parameters the model output metric. In addition, we also approximate true risk by using the Chebyshev' inequality. Numerical results are presented and compared to the corresponding Monte Carlo simulations ones.The law of the iterated logarithm for a piecewise deterministic Markov process assured by the properties of the Markov chain given by its post-jump locationshttps://zbmath.org/1472.601252021-11-25T18:46:10.358925Z"Czapla, Dawid"https://zbmath.org/authors/?q=ai:czapla.dawid"Hille, Sander C."https://zbmath.org/authors/?q=ai:hille.sander-cornelis"Horbacz, Katarzyna"https://zbmath.org/authors/?q=ai:horbacz.katarzyna"Wojewódka-Ściążko, Hanna"https://zbmath.org/authors/?q=ai:wojewodka-sciazko.hannaSummary: In the paper, we consider some piecewise deterministic Markov process, whose continuous component evolves according to semiflows, which are switched at the jump times of a Poisson process. The associated Markov chain describes the states of this process directly after the jumps. Certain ergodic properties of these two dynamical systems have been already investigated in our recent papers. We now aim to establish the law of the iterated logarithm for the aforementioned continuous-time process. Moreover, we intend to do this using the already proven properties of the discrete-time system. The abstract model under consideration has interesting interpretations in real-life sciences, such as biology. Among others, it can be used to describe the stochastic dynamics of gene expression.Mixing of the square plaquette model on a critical length scalehttps://zbmath.org/1472.601262021-11-25T18:46:10.358925Z"Chleboun, Paul"https://zbmath.org/authors/?q=ai:chleboun.paul"Smith, Aaron"https://zbmath.org/authors/?q=ai:smith.aaron-carl|smith.aaron-mAuthors' abstract: Plaquette models are short range ferromagnetic spin models that play a key role in the dynamic facilitation approach to the liquid glass transition. In this paper we study the dynamics of the square plaquette model at the smallest of the three critical length scales discovered in [the first author et al., J. Stat. Phys. 169, No. 3, 441--471 (2017; Zbl 1382.82009)]. Our main results are estimates of the spectral gap and mixing time for two natural boundary conditions. As a consequence, we observe that these time scales depend heavily on the boundary condition in this scaling regime.Mixing time and cutoff for the weakly asymmetric simple exclusion processhttps://zbmath.org/1472.601272021-11-25T18:46:10.358925Z"Labbé, Cyril"https://zbmath.org/authors/?q=ai:labbe.cyril"Lacoin, Hubert"https://zbmath.org/authors/?q=ai:lacoin.hubertSummary: We consider the simple exclusion process with \(k\) particles on a segment of length \(N\) performing random walks with transition \(p > 1/2\) to the right and \(q = 1 - p\) to the left. We focus on the case where the asymmetry in the jump rates \(b = p - q > 0\) vanishes in the limit when \(N\) and \(k\) tend to infinity, and obtain sharp asymptotics for the mixing times of this sequence of Markov chains in the two cases where the asymmetry is either much larger or much smaller than \((\log k)/N\). We show that in the former case \((b \gg (\log k)/N)\), the mixing time corresponds to the time needed to reach macroscopic equilibrium, like for the strongly asymmetric (i.e., constant \(b)\) case studied in [the authors, Ann. Probab. 47, No. 3, 1541--1586 (2019; Zbl 1466.60152)], while the latter case \((b\ll (\log k)/N)\) macroscopic equilibrium is not sufficient for mixing and one must wait till local fluctuations equilibrate, similarly to what happens in the symmetric case worked out in
[the second author, Ann. Probab. 44, No. 2, 1426--1487 (2016; Zbl 1408.60061)]. In both cases, convergence to equilibrium is abrupt: we have a cutoff phenomenon for the total-variation distance. We present a conjecture for the remaining regime when the asymmetry is of order \((\log k)/N\).Down/up crossing properties of weighted Markov collision processeshttps://zbmath.org/1472.601282021-11-25T18:46:10.358925Z"Li, Yanyun"https://zbmath.org/authors/?q=ai:li.yanyun"Li, Junping"https://zbmath.org/authors/?q=ai:li.junpingThe current paper focuses on weighted Markov collision processes, shortly WMC, which are continuous-time Markov chains with state space \(\mathbb{Z}_{+}\) and transition matrix defined in terms of births rates \((b_j)_{j\geq 0}\) and weights \((w_j)_{j\geq 0}\). For WMC, the authors investigate down/up crossing numbers. More precisely, in the main result Theorem 3.2 the joint probability generating function of down crossing and up crossing numbers for WMC is obtained. For making the understanding easier, several examples are considered towards the end of the work.Geometric fluid approximation for general continuous-time Markov chainshttps://zbmath.org/1472.601292021-11-25T18:46:10.358925Z"Michaelides, Michalis"https://zbmath.org/authors/?q=ai:michaelides.michalis-p"Hillston, Jane"https://zbmath.org/authors/?q=ai:hillston.jane"Sanguinetti, Guido"https://zbmath.org/authors/?q=ai:sanguinetti.guidoSummary: Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular population structure which suggests a natural continuous state-space endowed with a dynamics for the approximating process. We construct here a general method based on spectral analysis of the transition matrix of the CTMC, without the need for a population structure. Specifically, we use the popular manifold learning method of diffusion maps to analyse the transition matrix as the operator of a hidden continuous process. An embedding of states in a continuous space is recovered, and the space is endowed with a drift vector field inferred via Gaussian process regression. In this manner, we construct an ordinary differential equation whose solution approximates the evolution of the CTMC mean, mapped onto the continuous space (known as the fluid limit).On three methods for bounding the rate of convergence for some continuous-time Markov chainshttps://zbmath.org/1472.601302021-11-25T18:46:10.358925Z"Zeifman, Alexander"https://zbmath.org/authors/?q=ai:zeifman.alexander-i"Satin, Yacov"https://zbmath.org/authors/?q=ai:satin.yacov"Kryukova, Anastasia"https://zbmath.org/authors/?q=ai:kryukova.anastasia"Razumchik, Rostislav"https://zbmath.org/authors/?q=ai:razumchik.rostislav-v"Kiseleva, Ksenia"https://zbmath.org/authors/?q=ai:kiseleva.ksenia"Shilova, Galina"https://zbmath.org/authors/?q=ai:shilova.galinaSummary: Consideration is given to three different analytical methods for the computation of upper bounds for the rate of convergence to the limiting regime of one specific class of (in)homogeneous continuous-time Markov chains. This class is particularly well suited to describe evolutions of the total number of customers in (in)homogeneous \(M/M/S\) queueing systems with possibly state-dependent arrival and service intensities, batch arrivals and services. One of the methods is based on the logarithmic norm of a linear operator function; the other two rely on Lyapunov functions and differential inequalities, respectively. Less restrictive conditions (compared with those known from the literature) under which the methods are applicable are being formulated. Two numerical examples are given. It is also shown that, for homogeneous birth-death Markov processes defined on a finite state space with all transition rates being positive, all methods yield the same sharp upper bound.Second time scale of the metastability of reversible inclusion processeshttps://zbmath.org/1472.601312021-11-25T18:46:10.358925Z"Kim, Seonwoo"https://zbmath.org/authors/?q=ai:kim.seonwooSummary: We investigate the \textit{second time scale} of the metastable behavior of the reversible inclusion process in an extension of the study by \textit{A. Bianchi} et al. [Electron. J. Probab. 22, Paper No. 70, 34 p. (2017; Zbl 1386.60319)], which presented the first time scale of the same model and conjectured the scheme of multiple time scales. We show that \(N/d_N^2\) is indeed the correct second time scale for the most general class of reversible inclusion processes, and thus prove the first conjecture of the foresaid study. Here, \(N\) denotes the number of particles, and \(d_N\) denotes the small scale of randomness of the system. The main obstacles of this research arise in \textit{calculating the sharp asymptotics for the capacities}, and in the fact that the methods employed in the former study are not directly applicable due to the complex geometry of particle configurations. To overcome these problems, we first \textit{thoroughly examine the landscape of the transition rates} to obtain a proper test function of the equilibrium potential, which provides the upper bound for the capacities. Then, we \textit{modify the induced test flow} and \textit{precisely estimate the equilibrium potential near the metastable valleys} to obtain the correct lower bound for the capacities.Littlewood-Paley-Stein estimates for non-local Dirichlet formshttps://zbmath.org/1472.601322021-11-25T18:46:10.358925Z"Li, Huaiqian"https://zbmath.org/authors/?q=ai:li.huaiqian"Wang, Jian"https://zbmath.org/authors/?q=ai:wang.jian.2Summary: We obtain the boundedness in \(L^p\) spaces for all \(1<p<\infty\) of the so-called vertical Littlewood-Paley functions for non-local Dirichlet forms in the metric measure space under some mild assumptions. For \(1<p\leq 2\), the pseudo-gradient is introduced to overcome the difficulty that chain rules are not available for non-local operators, and then the Mosco convergence is used to pave the way from the uniformly bounded jumping kernel case to the general case, while for \(2\leq p\leq\infty\), the Burkholder-Davis-Gundy inequality is effectively applied. The former method is analytic and the latter one is probabilistic. The results extend those for pure jump symmetric Lévy processes in Euclidean spaces.Transience and recurrence of Markov processes with constrained local timehttps://zbmath.org/1472.601332021-11-25T18:46:10.358925Z"Barker, Adam"https://zbmath.org/authors/?q=ai:barker.adamSummary: We study Markov processes conditioned so that their local time must grow slower than a prescribed function. Building upon recent work on Brownian motion with constrained local time in [\textit{I. Benjamini} and \textit{N. Berestycki}, Ann. Inst. Henri Poincaré, Probab. Stat. 47, No. 2, 539--558 (2011; Zbl 1216.60028); \textit{M. Kolb} and \textit{M. Savov}, Ann. Probab. 44, No. 6, 4083--4132 (2016; Zbl 1364.60095)], we study transience and recurrence for a broad class of Markov processes. In order to understand the local time, we determine the distribution of a nondecreasing Lévy process (the inverse local time) conditioned to remain above a given level which varies in time. We study a time-dependent region, in contrast to previous works in which a process is conditioned to remain in a fixed region (e.g. [\textit{D. Denisov} and \textit{V. Wachtel}, ibid. 43, No. 3, 992--1044 (2015; Zbl 1332.60066); \textit{R. Garbit}, J. Math. Kyoto Univ. 49, No. 3, 573--592 (2009; Zbl 1192.60091)]), so we must study boundary crossing probabilities for a family of curves, and thus obtain uniform asymptotics for such a family. Main results include necessary and sufficient conditions for transience or recurrence of the conditioned Markov process. We will explicitly determine the distribution of the inverse local time for the conditioned process, and in the transient case, we explicitly determine the law of the conditioned Markov process. In the recurrent case, we characterise the ``entropic repulsion envelope'' via necessary and sufficient conditions.Optimality of threshold stopping times for diffusion processeshttps://zbmath.org/1472.601342021-11-25T18:46:10.358925Z"Arkin, V. I."https://zbmath.org/authors/?q=ai:arkin.vadim-iParameter and dimension dependence of convergence rates to stationarity for reflecting Brownian motionshttps://zbmath.org/1472.601352021-11-25T18:46:10.358925Z"Banerjee, Sayan"https://zbmath.org/authors/?q=ai:banerjee.sayan"Budhiraja, Amarjit"https://zbmath.org/authors/?q=ai:budhiraja.amarjit-sSummary: We obtain rates of convergence to stationarity in \(L^1\)-Wasserstein distance for a \(d\)-dimensional reflected Brownian motion (RBM) in the nonnegative orthant that are explicit in the dimension and the system parameters. The results are then applied to a class of RBMs considered in [\textit{J. Blanchet} and \textit{C. Xinyun}, ``Rates of convergence to stationarity for multidimensional'', Preprint, \url{arXiv:1601.04111}] and to rank-based diffusions including the Atlas model. In both cases, we obtain explicit rates and bounds on relaxation times. In the first case we improve the relaxation time estimates of \(O(d^4 (\log d)^2)\) obtained in [loc. cit.] to \(O ((\log d)^2)\). In the latter case, we give the first results on explicit parameter and dimension dependent rates under the Wasserstein distance. The proofs do not require an explicit form for the stationary measure or reversibility of the process with respect to this measure, and cover settings where these properties are not available. In the special case of the standard Atlas model [\textit{E. R. Fernholz}, Stochastic portfolio theory. New York, NY: Springer (2002; Zbl 1049.91067)], we obtain a bound on the relaxation time of \(O(d^6(\log d)^2)\).Stochastic approximation of quasi-stationary distributions for diffusion processes in a bounded domainhttps://zbmath.org/1472.601362021-11-25T18:46:10.358925Z"Benaïm, Michel"https://zbmath.org/authors/?q=ai:benaim.michel"Champagnat, Nicolas"https://zbmath.org/authors/?q=ai:champagnat.nicolas"Villemonais, Denis"https://zbmath.org/authors/?q=ai:villemonais.denisThe paper studies a random process with reinforcement, which evolves following the dynamics of a given diffusion process in a bounded domain and is resampled according to its occupation measure when it reaches the boundary. They show that its occupation measure converges to the unique quasi-stationary distribution of the diffusion process absorbed at the boundary of the domain. Section 2 formulates the main assumptions and results. Section 3 contains useful general results on quasi-stationary distributions and proofs of new general results on the Green operator \(A\) which has its own interest. Section 4 is devoted to the proof the main result, which consists in checking that the occupation measure of the resampling points is (up to a time change and linearization) an asymptotic pseudo-trajectory of a measure-valued dynamical system related to the operator \(A\).Diffusions interacting through a random matrix: universality via stochastic Taylor expansionhttps://zbmath.org/1472.601372021-11-25T18:46:10.358925Z"Dembo, Amir"https://zbmath.org/authors/?q=ai:dembo.amir"Gheissari, Reza"https://zbmath.org/authors/?q=ai:gheissari.rezaSummary: Consider \((X_i(t))\) solving a system of \(N\) stochastic differential equations interacting through a random matrix \({\mathbf{J}} = (J_{ij})\) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of \((X_i(t))\), initialized from some \(\mu\) independent of \({\mathbf{J}} \), are universal, i.e., only depend on the choice of the distribution \(\mathbf{J}\) through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.Convergence, boundedness, and ergodicity of regime-switching diffusion processes with infinite memoryhttps://zbmath.org/1472.601382021-11-25T18:46:10.358925Z"Li, Jun"https://zbmath.org/authors/?q=ai:li.jun.10|li.jun.13|li.jun.1|li.jun.3|li.jun.14|li.jun.11|li.jun.12|li.jun.7|li.jun.8|li.jun|li.jun.6|li.jun.2"Xi, Fubao"https://zbmath.org/authors/?q=ai:xi.fubaoSummary: We study a class of diffusion processes, which are determined by solutions \(X(t)\) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain \(\Lambda (t)\). Under suitable conditions, we investigate convergence and boundedness of both the solutions \(X(t)\) and the functional solutions \(X_t\). We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process \((X_t, \Lambda (t))\), and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.On the finiteness of moments of the exit time of planar Brownian motion from comb domainshttps://zbmath.org/1472.601392021-11-25T18:46:10.358925Z"Boudabra, Maher"https://zbmath.org/authors/?q=ai:boudabra.maher"Markowsky, Greg"https://zbmath.org/authors/?q=ai:markowsky.greg-tSummary: A comb domain is defined to be the entire complex plain with a collection of vertical slits, symmetric over the real axis, removed. In this paper, we consider the question of determining whether the exit time of planar Brownian motion from such a domain has finite \(p\)-th moment. This question has been addressed before in relation to starlike domains, but these previous results do not apply to comb domains. Our main result is a sufficient condition on the location of the slits which ensures that the \(p\)-th moment of the exit time is finite. Several auxiliary results are also presented, including a construction of a comb domain whose exit time has infinite \(p\)-th moment for all \(p \geq 1/2\).Analysis of Markov jump processes under terminal constraintshttps://zbmath.org/1472.601402021-11-25T18:46:10.358925Z"Backenköhler, Michael"https://zbmath.org/authors/?q=ai:backenkohler.michael"Bortolussi, Luca"https://zbmath.org/authors/?q=ai:bortolussi.luca"Großmann, Gerrit"https://zbmath.org/authors/?q=ai:grossmann.gerrit"Wolf, Verena"https://zbmath.org/authors/?q=ai:wolf.verenaSummary: Many probabilistic inference problems such as stochastic filtering or the computation of rare event probabilities require model analysis under initial and terminal constraints. We propose a solution to this \textit{bridging problem} for the widely used class of population-structured Markov jump processes. The method is based on a state-space lumping scheme that aggregates states in a grid structure. The resulting approximate bridging distribution is used to iteratively refine relevant and truncate irrelevant parts of the state-space. This way, the algorithm learns a well-justified finite-state projection yielding guaranteed lower bounds for the system behavior under endpoint constraints. We demonstrate the method's applicability to a wide range of problems such as Bayesian inference and the analysis of rare events.
For the entire collection see [Zbl 1466.68015].The coalescent structure of continuous-time Galton-Watson treeshttps://zbmath.org/1472.601412021-11-25T18:46:10.358925Z"Harris, Simon C."https://zbmath.org/authors/?q=ai:harris.simon-c"Johnston, Samuel G. G."https://zbmath.org/authors/?q=ai:johnston.samuel-g-g"Roberts, Matthew Iain"https://zbmath.org/authors/?q=ai:roberts.matthew-iSummary: Take a continuous-time Galton-Watson tree. If the system survives until a large time \(T\), then choose \(k\) particles uniformly from those alive. What does the ancestral tree drawn out by these \(k\) particles look like? Some special cases are known but we give a more complete answer. We concentrate on near-critical cases where the mean number of offspring is \(1 + \mu/T\) for some \(\mu \in \mathbb{R}\), and show that a scaling limit exists as \(T \to \infty\). Viewed backwards in time, the resulting coalescent process is topologically equivalent to Kingman's coalescent, but the times of coalescence have an interesting and highly nontrivial structure. The randomly fluctuating population size, as opposed to constant size populations where the Kingman coalescent more usually arises, have a pronounced effect on both the results and the method of proof required. We give explicit formulas for the distribution of the coalescent times, as well as a construction of the genealogical tree involving a mixture of independent and identically distributed random variables. In general subcritical and supercritical cases it is not possible to give such explicit formulas, but we highlight the special case of birth-death processes.Extremum of a time-inhomogeneous branching random walkhttps://zbmath.org/1472.601422021-11-25T18:46:10.358925Z"Hou, Wanting"https://zbmath.org/authors/?q=ai:hou.wanting"Zhang, Xiaoyue"https://zbmath.org/authors/?q=ai:zhang.xiaoyue"Hong, Wenming"https://zbmath.org/authors/?q=ai:hong.wenmingSummary: Consider a time-inhomogeneous branching random walk, generated by the point process \(L_n\) which composed by two independent parts: `branching' offspring \(X_n\) with the mean \(1 + B(1 + n)^{- \beta }\) for \(\beta \in (0, 1)\) and `displacement' \( \xi_n\) with a drift \(A(1 + n)^{-2 \alpha }\) for \(\alpha \in (0, 1/2)\); where the `branching' process is supercritical for \(B > 0\) but `asymptotically critical' and the drift of the `displacement' \( \xi_n\) is strictly positive or negative for \(|A| > 0\) but `asymptotically' goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the `asymptotical' parameter \(\beta\) and \(\alpha \).Lower deviations for supercritical branching processes with immigrationhttps://zbmath.org/1472.601432021-11-25T18:46:10.358925Z"Sun, Qi"https://zbmath.org/authors/?q=ai:sun.qi"Zhang, Mei"https://zbmath.org/authors/?q=ai:zhang.meiSummary: For a supercritical branching processes with immigration \(\{Z_n\}\), it is known that under suitable conditions on the offspring and immigration distributions, \(Z_n / m^n\) converges almost surely to a finite and strictly positive limit, where \(m\) is the offspring mean. We are interested in the limiting properties of \(\mathbb{P} (Z_n = k_n )\) with \(k_n = o(m^n )\) as \(n \rightarrow \infty \). We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.Moments of first hitting times for birth-death processes on treeshttps://zbmath.org/1472.601442021-11-25T18:46:10.358925Z"Zhang, Yuhui"https://zbmath.org/authors/?q=ai:zhang.yuhuiSummary: An explicit and recursive representation is presented for moments of the first hitting times of birth-death processes on trees. Based on that, the criteria on ergodicity, strong ergodicity, and \(\ell \)-ergodicity of the processes as well as a necessary condition for exponential ergodicity are obtained.Wald's martingale and the conditional distributions of absorption time in the Moran processhttps://zbmath.org/1472.601452021-11-25T18:46:10.358925Z"Monk, Travis"https://zbmath.org/authors/?q=ai:monk.travis"van Schaik, André"https://zbmath.org/authors/?q=ai:van-schaik.andreSummary: Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moranbirth-death process, where that quantity is the number of mutants in a population. In such processes, we are often interested in their absorption (i.e. fixation) probabilities and the conditional distributions of absorption time. Those conditional time distributions can be very difficult to calculate, even for relatively simple processes like the Moran birth-death model. Instead of considering the time to absorption, we consider a closely related quantity: the number of mutant population size changes before absorption. We use Wald's martingale to obtain the conditional characteristic functions of that quantity in the Moran process. Our expressions are novel, analytical and exact, and their parameter dependence is explicit. We use our results to approximate the conditional characteristic functions of absorption time. We state the conditions under which that approximation is particularly accurate. Martingales are an elegant framework to solve principal problems of evolutionary stochastic processes. They do not require us to evaluate recursion relations, so when they are applicable, we can quickly and tractably obtain absorption probabilities and times of evolutionary models.Correction to: ``Wald's martingale and the conditional distributions of absorption time in the Moran process''https://zbmath.org/1472.601462021-11-25T18:46:10.358925Z"Monk, Travis"https://zbmath.org/authors/?q=ai:monk.travis"van Schaik, André"https://zbmath.org/authors/?q=ai:van-schaik.andreCorrection to the authors' paper [Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 476, No. 2241, Article ID 20200135, 20 p. (2020; Zbl 1472.60145)].Site frequency spectrum of the Bolthausen-Sznitman coalescenthttps://zbmath.org/1472.601472021-11-25T18:46:10.358925Z"Kersting, Götz"https://zbmath.org/authors/?q=ai:kersting.gotz-dietrich"Siri-Jégousse, Arno"https://zbmath.org/authors/?q=ai:siri-jegousse.arno"Wences, Alejandro H."https://zbmath.org/authors/?q=ai:wences.alejandro-hSummary: We derive explicit formulas for the two first moments of the site frequency spectrum \((SFS_{n,b})_{1 \leq b \leq n -1}\) of the Bolthausen-Sznitman coalescent along with some precise and efficient approximations, even for small sample sizes \(n\). These results provide new \(L^2\)-asymptotics for some values of \(b=o(n)\). We also study the length of internal branches carrying \(b > n/2\) individuals, we provide their joint distribution function as well as a convergence in law for their marginal distribution. Our results rely on the random recursive tree construction of the Bolthausen-Sznitman coalescent.On the exit time from open sets of some semi-Markov processeshttps://zbmath.org/1472.601482021-11-25T18:46:10.358925Z"Ascione, Giacomo"https://zbmath.org/authors/?q=ai:ascione.giacomo"Pirozzi, Enrica"https://zbmath.org/authors/?q=ai:pirozzi.enrica"Toaldo, Bruno"https://zbmath.org/authors/?q=ai:toaldo.brunoSummary: In this paper we characterize the distribution of the first exit time from an arbitrary open set for a class of semi-Markov processes obtained as time-changed Markov processes. We estimate the asymptotic behaviour of the survival function (for large \(t)\) and of the distribution function (for small \(t)\) and we provide some conditions for absolute continuity. We have been inspired by a problem of neurophyshiology and our results are particularly usefull in this field, precisely for the so-called Leaky Integrate-and-Fire (LIF) models: the use of semi-Markov processes in these models appear to be realistic under several aspects, for example, it makes the intertimes between spikes a r.v. with infinite expectation, which is a desiderable property. Hence, after the theoretical part, we provide a LIF model based on semi-Markov processes.Target competition for resources under multiple search-and-capture events with stochastic resettinghttps://zbmath.org/1472.601492021-11-25T18:46:10.358925Z"Bressloff, P. C."https://zbmath.org/authors/?q=ai:bressloff.paul-cSummary: We develop a general framework for analysing the distribution of resources in a population of targets under multiple independent search-and-capture events. Each event involves a single particle executing a stochastic search that resets to a fixed location \(x_{}r\) at a random sequence of times. Whenever the particle is captured by a target, it delivers a packet of resources and then returns to \(x_{}r\), where it is reloaded with cargo and a new round of search and capture begins. Using renewal theory, we determine the mean number of resources in each target as a function of the splitting probabilities and unconditional mean first passage times of the corresponding search process without resetting. We then use asymptotic PDE methods to determine the effects of resetting on the distribution of resources generated by diffusive search in a bounded two-dimensional domain with \(N\) small interior targets. We show that slow resetting increases the total number of resources \(M_{tot}\) across all targets provided that \(\sum_{j = 1}^N G( \text{x}_r, \text{x}_j) < 0\), where \(G\) is the Neumann Green's function and \(x_{}j\) is the location of the \(j\)-th target. This implies that \(M_{tot}\) can be optimized by varying \(r\). We also show that the \(k\)-th target has a competitive advantage if \(\sum_{j = 1}^N G( \text{x}_r, \text{x}_j) > N G( \text{x}_r, \text{x}_k)\).Spectral gap in mean-field \({\mathcal{O}}(n)\)-modelhttps://zbmath.org/1472.601502021-11-25T18:46:10.358925Z"Becker, Simon"https://zbmath.org/authors/?q=ai:becker.simon"Menegaki, Angeliki"https://zbmath.org/authors/?q=ai:menegaki.angelikiSummary: We study the dependence of the spectral gap for the generator of the Ginzburg-Landau dynamics for all \(\mathcal O(n)\)-\textit{models} with mean-field interaction and magnetic field, below and at the critical temperature on the number \(N\) of particles. For our analysis of the Gibbs measure, we use a one-step renormalization approach and semiclassical methods to study the eigenvalue-spacing of an auxiliary Schrödinger operator.The Schelling model on \(\mathbb{Z}\)https://zbmath.org/1472.601512021-11-25T18:46:10.358925Z"Deijfen, Maria"https://zbmath.org/authors/?q=ai:deijfen.maria"Vilkas, Timo"https://zbmath.org/authors/?q=ai:vilkas.timoSummary: A version of the Schelling model on \(\mathbb{Z}\) is defined, where two types of agents are allocated on the sites. An agent prefers to be surrounded by other agents of its own type, and may choose to move if this is not the case. It then sends a request to an agent of opposite type chosen according to some given moving distribution and, if the move is beneficial for both agents, they swap location. We show that certain choices in the dynamics are crucial for the properties of the model. In particular, the model exhibits different asymptotic behavior depending on whether the moving distribution has bounded or unbounded support. Furthermore, the behavior changes if the agents are lazy in the sense that they only swap location if this strictly improves their situation. Generalizations to a version that includes multiple types are discussed. The work provides a rigorous analysis of so called Kawasaki dynamics on an infinite structure with local interactions.Cover time for the frog model on treeshttps://zbmath.org/1472.601522021-11-25T18:46:10.358925Z"Hoffman, Christopher"https://zbmath.org/authors/?q=ai:hoffman.christopher"Johnson, Tobias"https://zbmath.org/authors/?q=ai:johnson.tobias"Junge, Matthew"https://zbmath.org/authors/?q=ai:junge.matthewSummary: The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of \(\mu\) on the full \(d\)-ary tree of height \(n\). If \(\mu=\Omega(d^2)\), all of the vertices are visited in time \(\Theta(n\log n)\) with high probability. Conversely, if \(\mu=O(d)\) the cover time is \(\exp (\Theta(\sqrt{n}))\) with high probability.Diffusive scaling of the Kob-Andersen model in \({\mathbb{Z}}^d \)https://zbmath.org/1472.601532021-11-25T18:46:10.358925Z"Martinelli, F."https://zbmath.org/authors/?q=ai:martinelli.fabio"Shapira, A."https://zbmath.org/authors/?q=ai:shapira.alon-z|shapira.aasaf|shapira.andrew|shapira.asaf|shapira.assaf"Toninelli, C."https://zbmath.org/authors/?q=ai:toninelli.cristinaSummary: We consider the Kob-Andersen model, a cooperative lattice gas with kinetic constraints which has been widely analysed in the physics literature in connection with the study of the liquid/glass transition. We consider the model in a finite box of linear size \(L\) with sources at the boundary. Our result, which holds in any dimension and significantly improves upon previous ones, establishes for any positive vacancy density \(q\) a purely diffusive scaling of the relaxation time \(T_{\text{rel}}(q,L)\) of the system. Furthermore, as \(q\downarrow0\) we prove upper and lower bounds on \(L^{-2}T_{\text{rel}}(q,L)\) which agree with the physicists belief that the dominant equilibration mechanism is a cooperative motion of rare large droplets of vacancies. The main tools combine a recent set of ideas and techniques developed to establish universality results for kinetically constrained spin models, with methods from bootstrap percolation, oriented percolation and canonical flows for Markov chains.Spread of an infection on the zero range processhttps://zbmath.org/1472.601542021-11-25T18:46:10.358925Z"Baldasso, Rangel"https://zbmath.org/authors/?q=ai:baldasso.rangel"Teixeira, Augusto"https://zbmath.org/authors/?q=ai:teixeira.augusto-quadrosSummary: We study the spread of an infection on top of a moving population. The environment evolves as a zero range process on the integer lattice starting in equilibrium. At time zero, the set of infected particles is composed by those which are on the negative axis, while particles at the right of the origin are considered healthy. A healthy particle immediately becomes infected if it shares a site with an infected particle. We prove that the front of the infection wave travels to the right with positive and finite velocity. As a central step in the proof of these results, we prove a space-time decoupling for the zero range process which is interesting on its own. Using a sprinkling technique, we derive an estimate on the correlation of functions of the space of trajectories whose supports are sufficiently far away.A limit theorem for the survival probability of a simple random walk among power-law renewal obstacleshttps://zbmath.org/1472.601552021-11-25T18:46:10.358925Z"Poisat, Julien"https://zbmath.org/authors/?q=ai:poisat.julien"Simenhaus, François"https://zbmath.org/authors/?q=ai:simenhaus.francoisSummary: We consider a one-dimensional simple random walk surviving among a field of static soft obstacles: each time it meets an obstacle the walk is killed with probability \(1 - e^{-\beta}\), where \(\beta\) is a positive and fixed parameter. The positions of the obstacles are sampled independently from the walk and according to a renewal process. The increments between consecutive obstacles, or gaps, are assumed to have a power-law decaying tail with exponent \(\gamma > 0\). We prove convergence in law for the properly rescaled logarithm of the quenched survival probability as time goes to infinity. The normalization exponent is \(\gamma/(\gamma+2)\), while the limiting law writes as a variational formula with both universal and nonuniversal features. The latter involves (i) a Poisson point process that emerges as the universal scaling limit of the properly rescaled gaps and (ii) a function of the parameter \(\beta\) that we call asymptotic cost of crossing per obstacle and that may, in principle, depend on the details of the gap distribution. Our proof suggests a confinement strategy of the walk in a single large gap. This model may also be seen as a \((1+1)\)-directed polymer among many repulsive interfaces, in which case \(\beta\) corresponds to the strength of repulsion, the survival probability to the partition function and its logarithm to the finite-volume free energy.Asymptotic behavior of the integrated density of states for random point fields associated with certain Fredholm determinantshttps://zbmath.org/1472.601562021-11-25T18:46:10.358925Z"Ueki, Naomasa"https://zbmath.org/authors/?q=ai:ueki.naomasaSummary: The asymptotic behavior of the integrated density of states of a Schrödinger operator with positive potentials located around all sample points of some random point field at the infimum of the spectrum is investigated. The random point field is taken from a subclass of the class given by \textit{T. Shirai} and \textit{Y. Takahashi} [J. Funct. Anal. 205, No. 2, 414--463 (2003; Zbl 1051.60052)] in terms of the Fredholm determinant. In the subclass, the obtained leading orders are the same as the well known results for the Poisson point fields, and the character of the random field appears in the leading constants. The random point fields associated with the sine kernel and the Ginibre random point field are well studied examples not included in the above subclass, though they are included in the class by Shirai and Takahashi. By applying the results on asymptotics of the hole probability for these random fields, the corresponding asymptotic behaviors of the densities of states are also investigated in the case where the single site potentials have compact supports. The same method also applies to another well studied example, the zeros of a Gaussian random analytic function.Non-explosion criteria for rough differential equations driven by unbounded vector fieldshttps://zbmath.org/1472.601572021-11-25T18:46:10.358925Z"Bailleul, Ismael"https://zbmath.org/authors/?q=ai:bailleul.ismael-f"Catellier, Remi"https://zbmath.org/authors/?q=ai:catellier.remiSummary: We give in this note a simple treatment of the non-explosion problem for rough differential equations driven by unbounded vector fields and weak geometric rough paths of arbitrary roughness.WITHDRAWN: ``Cumulative residual extropy of minimum ranked set sampling with unequal samples''https://zbmath.org/1472.620102021-11-25T18:46:10.358925Z"Kazemi, Mohammad Reza"https://zbmath.org/authors/?q=ai:kazemi.mohammad-reza"Tahmasebi, Saeid"https://zbmath.org/authors/?q=ai:tahmasebi.saeid"Calì, Camilla"https://zbmath.org/authors/?q=ai:cali.camilla"Longobardi, Maria"https://zbmath.org/authors/?q=ai:longobardi.mariaFrom the text: The Publisher regrets that this article is an accidental duplication of an article that has already been published [the authors, ibid. 10, Article ID 100156, 11 p. (2021; Zbl 1470.62027)]. The duplicate article has therefore been withdrawn.Consistency of empirical Bayes and kernel flow for hierarchical parameter estimationhttps://zbmath.org/1472.620122021-11-25T18:46:10.358925Z"Chen, Yifan"https://zbmath.org/authors/?q=ai:chen.yifan"Owhadi, Houman"https://zbmath.org/authors/?q=ai:owhadi.houman"Stuart, Andrew M."https://zbmath.org/authors/?q=ai:stuart.andrew-mSummary: Gaussian process regression has proven very powerful in statistics, machine learning and inverse problems. A crucial aspect of the success of this methodology, in a wide range of applications to complex and real-world problems, is hierarchical modeling and learning of hyperparameters. The purpose of this paper is to study two paradigms of learning hierarchical parameters: one is from the probabilistic Bayesian perspective, in particular, the empirical Bayes approach that has been largely used in Bayesian statistics; the other is from the deterministic and approximation theoretic view, and in particular the kernel flow algorithm that was proposed recently in the machine learning literature. Analysis of their consistency in the large data limit, as well as explicit identification of their implicit bias in parameter learning, are established in this paper for a Matérn-like model on the torus. A particular technical challenge we overcome is the learning of the regularity parameter in the Matérn-like field, for which consistency results have been very scarce in the spatial statistics literature. Moreover, we conduct extensive numerical experiments beyond the Matérn-like model, comparing the two algorithms further. These experiments demonstrate learning of other hierarchical parameters, such as amplitude and lengthscale; they also illustrate the setting of model misspecification in which the kernel flow approach could show superior performance to the more traditional empirical Bayes approach.Safe adaptive importance sampling: a mixture approachhttps://zbmath.org/1472.620162021-11-25T18:46:10.358925Z"Delyon, Bernard"https://zbmath.org/authors/?q=ai:delyon.bernard"Portier, François"https://zbmath.org/authors/?q=ai:portier.francoisAdaptive importance sampling (AIS) constitutes new samples, such as particles in statistical physics, generated under certain probability distribution called policy \(q_k\) and the next policy \(q_{k+1}\) uses the new particles adaptively. In the earlier works, the policy is chosen as the kernel density estimate based on the previous particles reweighted by their importance weights. The authors propose a new approach called `safe adaptive importance sampling' (SAIS) which estimates the policy as a mixture of kernel density estimate and certain `safe' density with heavier tails. They also consider the functional approximation and derive convergence rates, leading to a central limit theorem for the estimates. It is observed that the asymptotic variance with this procedure is the same as that of an `oracle' procedure. Further, a subsampling approach can be adopted to reduce the computational time involved without loosing the original efficiency. A simulation study at the end illustrates the practical nature of the algorithms developed. A section at the end gives detailed mathematical proofs including two appendices. There is a rich list of useful references.The KLR-theorem revisitedhttps://zbmath.org/1472.620272021-11-25T18:46:10.358925Z"Kagan, Abram"https://zbmath.org/authors/?q=ai:kagan.abram-meerovichSummary: For independent random variables \(X_1,\dots,X_n;Y_1,\dots,Y_n\) with all \(X_i\) identically distributed and same for \(Y_j\), we study the relation
\[
E\{a\bar{X}+b\bar{Y}|X_1-\bar{X}+Y_1-\bar{Y},\dots,X_n-\bar{X}+Y_n-\bar{Y}\}=\mathrm{const}
\]
with \(a,b\) some constants. It is proved that for \(n\geq 3\) and \(ab>0\) the relation holds iff \(X_i\) and \(Y_j\) are Gaussian.
A new characterization arises in case of \(a=1\), \(b=-1\). In this case either \(X_i\) or \(Y_j\) or both have a Gaussian component. It is the first (at least known to the author) case when presence of a Gaussian component is a characteristic property. For the KLR-theorem see [Zbl 0168.40203)].Approximations to weighted sums of random variableshttps://zbmath.org/1472.620302021-11-25T18:46:10.358925Z"Kumar, Amit N."https://zbmath.org/authors/?q=ai:kumar.amit-nSummary: In this paper, we obtain error bound for pseudo-binomial and negative binomial approximations to weighted sums of locally dependent random variables, using Stein's method. We also discuss approximation results for weighted sums of independent random variables. We demonstrate our results through some applications in finance and runs in statistics.Quantification of model uncertainty on path-space via goal-oriented relative entropyhttps://zbmath.org/1472.620422021-11-25T18:46:10.358925Z"Birrell, Jeremiah"https://zbmath.org/authors/?q=ai:birrell.jeremiah"Katsoulakis, Markos A."https://zbmath.org/authors/?q=ai:katsoulakis.markos-a"Rey-Bellet, Luc"https://zbmath.org/authors/?q=ai:rey-bellet.lucSummary: Quantifying the impact of parametric and model-form uncertainty on the predictions of stochastic models is a key challenge in many applications. Previous work has shown that the relative entropy rate is an effective tool for deriving path-space uncertainty quantification (UQ) bounds on ergodic averages. In this work we identify appropriate information-theoretic objects for a wider range of quantities of interest on path-space, such as hitting times and exponentially discounted observables, and develop the corresponding UQ bounds. In addition, our method yields tighter UQ bounds, even in cases where previous relative-entropy-based methods also apply, \textit{e.g.}, for ergodic averages. We illustrate these results with examples from option pricing, non-reversible diffusion processes, stochastic control, semi-Markov queueing models, and expectations and distributions of hitting times.The law of the iterated logarithm and maximal smoothing principle for the kernel distribution function estimatorhttps://zbmath.org/1472.620482021-11-25T18:46:10.358925Z"Swanepoel, Jan W. H."https://zbmath.org/authors/?q=ai:swanepoel.jan-w-hSummary: Two new properties of the kernel distribution function estimator of diverse nature are derived. Firstly, a law of the iterated logarithm is proved for both the integrated absolute error and the integrated squared error of the estimator. Secondly, the maximal smoothing principle in kernel density estimation developed by Terrell is extended to kernel distribution function estimation, which allows, among others, the derivation of an alternative quick-and-simple bandwidth selector. In fact, there is a common link between the two topics: both problems are solved through the use of the same, not-so-standard, methodology. The results based on simulated data and a real data set are also presented.Erratum to: ``Estimation and uncertainty quantification for extreme quantile regions''https://zbmath.org/1472.620582021-11-25T18:46:10.358925Z"Beranger, Boris"https://zbmath.org/authors/?q=ai:beranger.boris"Padoan, Simone A."https://zbmath.org/authors/?q=ai:padoan.simone-a"Sisson, Scott A."https://zbmath.org/authors/?q=ai:sisson.scott-aErratum to the authors' paper [ibid. 24, No. 2, 349--375 (2021; Zbl 1466.62291)].Some asymptotic properties of conditional density function for functional data under random censorshiphttps://zbmath.org/1472.620652021-11-25T18:46:10.358925Z"Akkal, Fatima"https://zbmath.org/authors/?q=ai:akkal.fatima"Rabhi, Abbes"https://zbmath.org/authors/?q=ai:rabhi.abbes"Keddani, Latifa"https://zbmath.org/authors/?q=ai:keddani.latifaSummary: In this work, we investigate the asymptotic properties of a nonparametric mode of a conditional density when the real response variable is censored and the explanatory variable is valued in a semi-metric space under ergodic data. First of all, we establish asymptotic properties for a conditional density estimator from which we derive an central limit theorem (CLT) of the conditional mode estimator. Simulation study is also presented to illustrate the validity and finite sample performance of the considered estimator.On the uniform-in-bandwidth consistency of the general conditional \(U\)-statistics based on the copula representationhttps://zbmath.org/1472.620672021-11-25T18:46:10.358925Z"Bouzebda, Salim"https://zbmath.org/authors/?q=ai:bouzebda.salim"Elhattab, Issam"https://zbmath.org/authors/?q=ai:elhattab.issam"Nemouchi, Boutheina"https://zbmath.org/authors/?q=ai:nemouchi.boutheinaSummary: \textit{W. Stute} [Ann. Probab. 19, No. 2, 812--825 (1991; Zbl 0770.60035)] introduced a class of estimators called conditional \(U\)-statistics of
\[
\mathbb{E} (\varphi (Y_1 , \ldots , Y_m ) \, | \, (X_1 , \ldots , X_m ) = \mathbf{t} ), \quad \text{for } \mathbf{t} \in \mathbb{R}^m .
\]
In the present work, we provide a new class of estimators of conditional \(U\)-statistics. More precisely, we investigate the conditional \(U\)-statistics based on copula representation. We establish the uniform-in-bandwidth consistency for the proposed estimator. Our theorems allow data-driven local bandwidths for these statistics. The theoretical uniform consistency results, established in this paper, are (or will be) key tools for many further developments in copula regression analysis. The performance of these procedures is evaluated through simulation in the context of the conditional Kendall's tau.A variance shift model for detection of outliers in the linear measurement error modelhttps://zbmath.org/1472.620702021-11-25T18:46:10.358925Z"Babadi, Babak"https://zbmath.org/authors/?q=ai:babadi.babak"Rasekh, Abdolrahman"https://zbmath.org/authors/?q=ai:rasekh.abdolrahman"Rasekhi, Ali Akbar"https://zbmath.org/authors/?q=ai:rasekhi.ali-akbar"Zare, Karim"https://zbmath.org/authors/?q=ai:zare.karim"Zadkarami, Mohammad Reza"https://zbmath.org/authors/?q=ai:zadkarami.mohammad-rezaSummary: We present a variance shift model for a linear measurement error model using the corrected likelihood of \textit{T. Nakamura} [Biometrika 77, No. 1, 127--137 (1990; Zbl 0691.62066)]. This model assumes that a single outlier arises from an observation with inflated variance. The corrected likelihood ratio and the score test statistics are proposed to determine whether the \(i\)th observation has an inflated variance. A parametric bootstrap procedure is used to obtain empirical distributions of the test statistics and a simulation study has been used to show the performance of proposed tests. Finally, a real data example is given for illustration.New families of bivariate copulas via unit Weibull distortionhttps://zbmath.org/1472.620712021-11-25T18:46:10.358925Z"Aldhufairi, Fadal A. A."https://zbmath.org/authors/?q=ai:aldhufairi.fadal-a-a"Sepanski, Jungsywan H."https://zbmath.org/authors/?q=ai:sepanski.jungsywan-hwangSummary: This paper introduces a new family of bivariate copulas constructed using a unit Weibull distortion. Existing copulas play the role of the base or initial copulas that are transformed or distorted into a new family of copulas with additional parameters, allowing more flexibility and better fit to data. We present a general form for the new bivariate copula function and its conditional and density distributions. The tail behaviors are investigated and indicate the unit Weibull distortion may result in new copulas with upper tail dependence when the base copula has no upper tail dependence. The concordance ordering and Kendall's tau are derived for the cases when the base copulas are Archimedean, such as the Clayton and Frank copulas. The Loss-ALEA data are analyzed to evaluate the performance of the proposed new families of copulas.Construction of copulas with hairpin supporthttps://zbmath.org/1472.620722021-11-25T18:46:10.358925Z"Chamizo, Fernando"https://zbmath.org/authors/?q=ai:chamizo-lorente.fernando"Fernández-Sánchez, Juan"https://zbmath.org/authors/?q=ai:fernandez-sanchez.juan"Úbeda-Flores, Manuel"https://zbmath.org/authors/?q=ai:ubeda-flores.manuelSummary: Consider a nondecreasing homeomorphisms \(f\) defined on \([0, 1]\) such that \(f(x)<x\) for all \(x\in ]0,1[\). In this paper, we provide necessary and sufficient conditions for such \(f\) to be part of a \(\mathcal{C}\)-hairpin that concentrates the mass of a bivariate copula. In addition, we study when copulas of this kind come from modular functions. Finally, under certain conditions, we give a multidimensional method that generalizes the bivariate case and allows to construct extreme points in the set of multidimensional copulas.Bimatrix variate Kummer-gamma distributionhttps://zbmath.org/1472.620752021-11-25T18:46:10.358925Z"Nagar, Daya K."https://zbmath.org/authors/?q=ai:nagar.daya-k"Roldán-Correa, Alejandro"https://zbmath.org/authors/?q=ai:roldan-correa.alejandro"Gupta, Arjun K."https://zbmath.org/authors/?q=ai:gupta.arjun-kSummary: In this article, we propose a bimatrix variate Kummer-gamma distribution. Several properties of this distribution including moment generating function, marginal and conditional distributions, moments are also derived. These results are given in terms of special functions of matrix arguments.Uncovering causality from multivariate Hawkes integrated cumulantshttps://zbmath.org/1472.620762021-11-25T18:46:10.358925Z"Achab, Massil"https://zbmath.org/authors/?q=ai:achab.massil"Bacry, Emmanuel"https://zbmath.org/authors/?q=ai:bacry.emmanuel"Gaïffas, Stéphane"https://zbmath.org/authors/?q=ai:gaiffas.stephane"Mastromatteo, Iacopo"https://zbmath.org/authors/?q=ai:mastromatteo.iacopo"Muzy, Jean-François"https://zbmath.org/authors/?q=ai:muzy.jean-francoisSummary: We design a new nonparametric method that allows one to estimate the matrix of integrated kernels of a multivariate Hawkes process. This matrix not only encodes the mutual influences of each node of the process, but also disentangles the causality relationships between them. Our approach is the first that leads to an estimation of this matrix \textit{without any parametric modeling and estimation of the kernels themselves}. As a consequence, it can give an estimation of causality relationships between nodes (or users), based on their activity timestamps (on a social network for instance), without knowing or estimating the shape of the activities lifetime. For that purpose, we introduce a moment matching method that fits the second-order and the third-order integrated cumulants of the process. A theoretical analysis allows us to prove that this new estimation technique is consistent. Moreover, we show, on numerical experiments, that our approach is indeed very robust with respect to the shape of the
kernels and
gives appealing results on the MemeTracker database and on financial order book data.Maximum likelihood drift estimation for a threshold diffusionhttps://zbmath.org/1472.620782021-11-25T18:46:10.358925Z"Lejay, Antoine"https://zbmath.org/authors/?q=ai:lejay.antoine"Pigato, Paolo"https://zbmath.org/authors/?q=ai:pigato.paoloThe drifted oscillating Brownian motion (DOBM) is an interesting process since it can be applied, specially, in Finance. For instance, at some conditions, the DOBM is a generalization of the Black-Scholes Model. The DOBM is function of the drift and diffusion parameters. In the frequentist context, the maximum likelihood (ML) method can be used to estimate the parameters of a model or process, however this procedure can not be always applied. Nevertheless, Lejay and Pigato derived the ML estimator for the drift parameters in the DOBM, for continuous and discrete observations. Additionally, they also evaluated the asymptotic properties of this estimator, which allowed them to derive confidence intervals.High-dimensional linear models: a random matrix perspectivehttps://zbmath.org/1472.620802021-11-25T18:46:10.358925Z"Namdari, Jamshid"https://zbmath.org/authors/?q=ai:namdari.jamshid"Paul, Debashis"https://zbmath.org/authors/?q=ai:paul.debashis"Wang, Lili"https://zbmath.org/authors/?q=ai:wang.lili.3Summary: Professor \textit{C. R. Rao}'s [Linear statistical inference and its applications. New York-London-Sydney: John Wiley and Sons, Inc. (1965; Zbl 0137.36203)] is a classic that has motivated several generations of statisticians in their pursuit of theoretical research. This paper looks into some of the fundamental problems associated with linear models, but in a scenario where the dimensionality of the observations is comparable to the sample size. This perspective, largely driven by contemporary advancements in random matrix theory, brings new insights and results that can be helpful even for solving relatively low-dimensional problems. This overview also brings into focus the fundamental roles played by the eigenvalues of large covariance-type matrices in the theory of high-dimensional multivariate statistics.Inference problem in generalized fractional Ornstein-Uhlenbeck processes with change-pointhttps://zbmath.org/1472.620812021-11-25T18:46:10.358925Z"Nkurunziza, Sévérien"https://zbmath.org/authors/?q=ai:nkurunziza.severienSummary: In this paper, we study an inference problem in generalized fractional Ornstein-Uhlenbeck (O-U) processes with an unknown change-point when the drift parameter is suspected to satisfy some constraints. The constraint considered is very general and, the testing problem studied generalizes a very recent inference problem in generalized O-U processes. We derive the unrestricted estimator (UE) and the restricted estimator (RE) and we establish the asymptotic properties of the UE and RE. We also propose some shrinkage-type estimators (SEs) as well as a test for testing the constraint. Finally, we derive the asymptotic power of the proposed test and we study the relative risk dominance of the proposed estimators.Estimating leverage scores via rank revealing methods and randomizationhttps://zbmath.org/1472.620822021-11-25T18:46:10.358925Z"Sobczyk, Aleksandros"https://zbmath.org/authors/?q=ai:sobczyk.aleksandros"Gallopoulos, Efstratios"https://zbmath.org/authors/?q=ai:gallopoulos.efstratiosModified martingale difference correlationshttps://zbmath.org/1472.620872021-11-25T18:46:10.358925Z"Zhou, Jingke"https://zbmath.org/authors/?q=ai:zhou.jingke"Zhu, Lixing"https://zbmath.org/authors/?q=ai:zhu.lixingSummary: To ameliorate some drawbacks of Martingale Difference Correlation (MDC) such as the asymmetry in the sense that for a pair of vectors, the value of \textit{MDC} may not be equal to 1, and the self-\textit{MDC} of any random vector can be different from vector to vector in value, we in this paper propose a modified MDC (MMDC). Further, as the corresponding partial MDC (PMDC), with controlling another random vector, cannot ensure the equivalence between conditional mean independence and zero PMDC, we then also propose a modified partial MDC (MPMDC) to guarantee, under some regularity conditions, the equivalence. We further investigate the theoretical properties of the corresponding unbiased estimators and apply them to variable screening and hypothesis testing. Numerical studies and real data analysis are conducted to examine their finite sample performances.Permutation entropy and bubble entropy: possible interactions and synergies between order and sorting relationshttps://zbmath.org/1472.620922021-11-25T18:46:10.358925Z"Cuesta-Frau, David"https://zbmath.org/authors/?q=ai:cuesta-frau.david"Vargas, Borja"https://zbmath.org/authors/?q=ai:vargas.borjaSummary: Despite its widely demonstrated usefulness, there is still room for improvement in the basic Permutation Entropy (PE) algorithm, as several subsequent studies have proposed in the recent years. For example, some improved PE variants try to address possible PE weaknesses, such as its only focus on ordinal information, and not on amplitude, or the possible detrimental impact of equal values in subsequences due to motif ambiguity. Other evolved PE methods try to reduce the influence of input parameters. A good representative of this last point is the Bubble Entropy (BE) method. BE is based on sorting relations instead of ordinal patterns, and its promising capabilities have not been extensively assessed yet. The objective of the present study was to comparatively assess the classification performance of this new method, and study and exploit the possible synergies between PE and BE. The claimed superior performance of BE over PE was first evaluated by conducting a series of time series classification tests over a varied and diverse experimental set. The results of this assessment apparently suggested that there is a complementary relationship between PE and BE, instead of a superior/inferior relationship. A second set of experiments using PE and BE simultaneously as the input features of a clustering algorithm, demonstrated that with a proper algorithm configuration, classification accuracy and robustness can benefit from both measures.Significance-based community detection in weighted networkshttps://zbmath.org/1472.621232021-11-25T18:46:10.358925Z"Palowitch, John"https://zbmath.org/authors/?q=ai:palowitch.john"Bhamidi, Shankar"https://zbmath.org/authors/?q=ai:bhamidi.shankar"Nobel, Andrew B."https://zbmath.org/authors/?q=ai:nobel.andrew-bSummary: Community detection is the process of grouping strongly connected nodes in a network. Many community detection methods for un-weighted networks have a theoretical basis in a null model. Communities discovered by these methods therefore have interpretations in terms of statistical significance. In this paper, we introduce a null for weighted networks called the continuous configuration model. First, we propose a community extraction algorithm for weighted networks which incorporates iterative hypothesis testing under the null. We prove a central limit theorem for edge-weight sums and asymptotic consistency of the algorithm under a weighted stochastic block model. We then incorporate the algorithm in a community detection method called CCME. To benchmark the method, we provide a simulation framework involving the null to plant ``background'' nodes in weighted networks with communities. We show that the empirical performance of CCME on these simulations is competitive with existing methods,
particularly
when overlapping communities and background nodes are present. To further validate the method, we present two real-world networks with potential background nodes and analyze them with CCME, yielding results that reveal macro-features of the corresponding systems.The role of differential equations in applied statisticshttps://zbmath.org/1472.621262021-11-25T18:46:10.358925Z"Kitsos, Christos P."https://zbmath.org/authors/?q=ai:kitsos.christos-p"Nisiotis, C. S. A."https://zbmath.org/authors/?q=ai:nisiotis.c-s-aSummary: The target of this paper is to discuss, investigate and present how the differential equations are applied in Statistics. The stochastic orientation of Statistics creates problems to adopt the differential equations as an individual tool, but Applied Statistics is using the differential equations either through applications from other fields, like Chemistry or as a tool to explain ``variation'' in stochastic processes.
For the entire collection see [Zbl 1471.34005].Statistical inference for stochastic differential equations with small noiseshttps://zbmath.org/1472.621322021-11-25T18:46:10.358925Z"Shen, Liang"https://zbmath.org/authors/?q=ai:shen.liang"Xu, Qingsong"https://zbmath.org/authors/?q=ai:xu.qingsongSummary: This paper proposes the least squares method to estimate the drift parameter for the stochastic differential equations driven by small noises, which is more general than pure jump \(\alpha\)-stable noises. The asymptotic property of this least squares estimator is studied under some regularity conditions. The asymptotic distribution of the estimator is shown to be the convolution of a stable distribution and a normal distribution, which is completely different from the classical cases.Statistical estimation of ergodic Markov chain kernel over discrete state spacehttps://zbmath.org/1472.621332021-11-25T18:46:10.358925Z"Wolfer, Geoffrey"https://zbmath.org/authors/?q=ai:wolfer.geoffrey"Kontorovich, Aryeh"https://zbmath.org/authors/?q=ai:kontorovich.leonid-aryehSummary: We investigate the statistical complexity of estimating the parameters of a discrete-state Markov chain kernel from a single long sequence of state observations. In the finite case, we characterize (modulo logarithmic factors) the minimax sample complexity of estimation with respect to the operator infinity norm, while in the countably infinite case, we analyze the problem with respect to a natural entry-wise norm derived from total variation. We show that in both cases, the sample complexity is governed by the mixing properties of the unknown chain, for which, in the finite-state case, there are known finite-sample estimators with fully empirical confidence intervals.Cointegration in high frequency datahttps://zbmath.org/1472.621342021-11-25T18:46:10.358925Z"Clinet, Simon"https://zbmath.org/authors/?q=ai:clinet.simon"Potiron, Yoann"https://zbmath.org/authors/?q=ai:potiron.yoannSummary: In this paper, we consider a framework adapting the notion of cointegration when two asset prices are generated by a driftless Itô-semimartingale featuring jumps with infinite activity, observed regularly and synchronously at high frequency. We develop a regression based estimation of the cointegrated relations method and show the related consistency and central limit theory when there is cointegration within that framework. We also provide a Dickey-Fuller type residual based test for the null of no cointegration against the alternative of cointegration, along with its limit theory. Under no cointegration, the asymptotic limit is the same as that of the original Dickey-Fuller residual based test, so that critical values can be easily tabulated in the same way. Finite sample indicates adequate size and good power properties in a variety of realistic configurations, outperforming original Dickey-Fuller and Phillips-Perron type residual based tests, whose sizes are distorted by non ergodic time-varying variance and power is altered by price jumps. Two empirical examples consolidate the Monte-Carlo evidence that the adapted tests can be rejected while the original tests are not, and vice versa.Asymptotic analysis of model selection criteria for general hidden Markov modelshttps://zbmath.org/1472.621382021-11-25T18:46:10.358925Z"Yonekura, Shouto"https://zbmath.org/authors/?q=ai:yonekura.shouto"Beskos, Alexandros"https://zbmath.org/authors/?q=ai:beskos.alexandros"Singh, Sumeetpal S."https://zbmath.org/authors/?q=ai:singh.sumeetpal-sSummary: The paper obtains analytical results for the asymptotic properties of Model Selection Criteria -- widely used in practice -- for a general family of hidden Markov models (HMMs), thereby substantially extending the related theory beyond typical `i.i.d.-like' model structures and filling in an important gap in the relevant literature. In particular, we look at the Bayesian and Akaike Information Criteria (BIC and AIC) and the model evidence. In the setting of nested classes of models, we prove that BIC and the evidence are strongly consistent for HMMs (under regularity conditions), whereas AIC is not weakly consistent. Numerical experiments support our theoretical results.Nonrigid registration using Gaussian processes and local likelihood estimationhttps://zbmath.org/1472.621412021-11-25T18:46:10.358925Z"Wiens, Ashton"https://zbmath.org/authors/?q=ai:wiens.ashton"Kleiber, William"https://zbmath.org/authors/?q=ai:kleiber.william"Nychka, Douglas"https://zbmath.org/authors/?q=ai:nychka.douglas-w"Barnhart, Katherine R."https://zbmath.org/authors/?q=ai:barnhart.katherine-rSummary: Surface registration, the task of aligning several multidimensional point sets, is a necessary task in many scientific fields. In this work, a novel statistical approach is developed to solve the problem of nonrigid registration. While the application of an affine transformation results in rigid registration, using a general nonlinear function to achieve nonrigid registration is necessary when the point sets require deformations that change over space. The use of a local likelihood-based approach using windowed Gaussian processes provides a flexible way to accurately estimate the nonrigid deformation. This strategy also makes registration of massive data sets feasible by splitting the data into many subsets. The estimation results yield spatially-varying local rigid registration parameters. Gaussian process surface models are then fit to the parameter fields, allowing prediction of the transformation parameters at unestimated locations, specifically at observation locations in the unregistered data set. Applying these transformations results in a global, nonrigid registration. A penalty on the transformation parameters is included in the likelihood objective function. Combined with smoothing of the local estimates from the surface models, the nonrigid registration model can prevent the problem of overfitting. The efficacy of the nonrigid registration method is tested in two simulation studies, varying the number of windows and number of points, as well as the type of deformation. The nonrigid method is applied to a pair of massive remote sensing elevation data sets exhibiting complex geological terrain, with improved accuracy and uncertainty quantification in a cross validation study versus two rigid registration methods.Relative stochastic orders of weighted frailty modelshttps://zbmath.org/1472.621462021-11-25T18:46:10.358925Z"He, Xu"https://zbmath.org/authors/?q=ai:he.xu"Xie, Hongmei"https://zbmath.org/authors/?q=ai:xie.hongmeiSummary: In this paper, relative stochastic comparisons of weighted frailty models with respect to relative hazard rate and relative mean residual life orders are considered. Some closure properties of the model in two relative stochastic orders sense are presented. Under some appropriate assumptions, we show how the variation of the frailty variable and the variation of the baseline variable translate into these two relative stochastic orders between the overall population variables.Dirichlet depths for point processhttps://zbmath.org/1472.621762021-11-25T18:46:10.358925Z"Qi, Kai"https://zbmath.org/authors/?q=ai:qi.kai"Chen, Yang"https://zbmath.org/authors/?q=ai:chen.yang.2|chen.yang.1"Wu, Wei"https://zbmath.org/authors/?q=ai:wu.wei|wu.wei.3|wu.wei.2|wu.wei.4Summary: Statistical depths have been well studied for multivariate and functional data over the past few decades, but remain under-explored for point processes. A first attempt on the notion of point process depth was conducted recently where the depth was defined as a weighted product of two terms: (1) the probability of the number of events in each process and (2) the depth of the event times conditioned on the number of events by using a Mahalanobis depth. We point out that multivariate depths such as the Mahalanobis depth cannot be directly used because they often neglect the important ordering property in the point process events. To deal with this problem, we propose a model-based approach for point process systematically. In particular, we develop a Dirichlet-distribution-based framework on the conditional depth term, where the new methods are referred to as Dirichlet depths. We examine mathematical properties of the new depths and conduct asymptotic analysis. In addition, we illustrate the new methods using various simulated and real experiment data. It is found that the proposed framework provides a reasonable center-outward rank and the new methods have accurate decoding in one neural spike train dataset.Persistent homology for low-complexity modelshttps://zbmath.org/1472.621802021-11-25T18:46:10.358925Z"Lotz, Martin"https://zbmath.org/authors/?q=ai:lotz.martinSummary: We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is determined by the Gaussian width of a structure associated with the dataset, rather than its size, and such a reduction can be computed efficiently. We further relate the Gaussian width to the doubling dimension of a finite metric space, which appears in the study of the complexity of other methods for approximating persistent homology. We can, therefore, literally replace the ambient dimension by an intrinsic notion of dimension related to the structure of the data.Flow-driven spectral chaos (FSC) method for long-time integration of second-order stochastic dynamical systemshttps://zbmath.org/1472.650122021-11-25T18:46:10.358925Z"Esquivel, Hugo"https://zbmath.org/authors/?q=ai:esquivel.hugo"Prakash, Arun"https://zbmath.org/authors/?q=ai:prakash.arun-j"Lin, Guang"https://zbmath.org/authors/?q=ai:lin.guangSummary: For decades, uncertainty quantification techniques based on the spectral approach have been demonstrated to be computationally more efficient than the Monte Carlo method for a wide variety of problems, particularly when the dimensionality of the probability space is relatively low. The time-dependent generalized polynomial chaos (TD-gPC) is one such technique that uses an evolving orthogonal basis to better represent the stochastic part of the solution space in time. In this paper, we present a new numerical method that uses the concept of \textit{enriched stochastic flow maps} to track the evolution of the stochastic part of the solution space in time. The computational cost of this proposed flow-driven stochastic chaos (FSC) method is an order of magnitude lower than TD-gPC for comparable solution accuracy. This gain in computational cost is realized because, unlike most existing methods, the number of basis vectors required to track the stochastic part of the solution space does not depend upon the dimensionality of the probability space. Four representative numerical examples are presented to demonstrate the performance of the FSC method for long-time integration of second-order stochastic dynamical systems in the context of stochastic dynamics of structures.Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integratorshttps://zbmath.org/1472.650132021-11-25T18:46:10.358925Z"Jin, Sixian"https://zbmath.org/authors/?q=ai:jin.sixian"Kobayashi, Kei"https://zbmath.org/authors/?q=ai:kobayashi.keiSummary: The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time change given by an inverse subordinator. SDEs to be considered are unique in two different aspects: (i) they contain two drift terms, one driven by the random time change and the other driven by a regular, non-random time variable; (ii) the standard Lipschitz assumption is replaced by that with a time-varying Lipschitz bound. The difficulty imposed by the first aspect is overcome via an approach that is significantly different from a well-known method based on the so-called duality principle. On the other hand, the second aspect requires the establishment of a criterion for the existence of exponential moments of functions of the random time change.Convergence of time-splitting approximations for degenerate convection-diffusion equations with a random sourcehttps://zbmath.org/1472.650992021-11-25T18:46:10.358925Z"Díaz-Adame, Roberto"https://zbmath.org/authors/?q=ai:diaz-adame.roberto"Jerez, Silvia"https://zbmath.org/authors/?q=ai:jerez.silviaIn this paper authors propose a time-splitting method for degenerate convectionś-diffusion equations perturbed stochastically by white noise. This work generalizes previous results on splitting operator techniques for stochastic hyperbolic conservation laws for the degenerate parabolic case. The convergence in \(L_{\mathrm{loc}}^p\) of the time-splitting operator scheme to the unique weak entropy solution is proven. Moreover, we analyze the performance of the splitting approximation by computing its convergence rate and showing numerical simulations for some benchmark examples, including a luid low application in porous media.Convergence analysis of the discrete duality finite volume scheme for the regularised Heston modelhttps://zbmath.org/1472.651062021-11-25T18:46:10.358925Z"Tibenský, Matúš"https://zbmath.org/authors/?q=ai:tibensky.matus"Handlovičová, Angela"https://zbmath.org/authors/?q=ai:handlovicova.angelaAuthors' abstract: The aim of the paper is to study the problem of financial derivatives pricing based on the idea of the Heston model introduced in [\textit{S. L. Heston}, Rev. Financ. Stud. 6, No. 2, 327--343 (1993; Zbl 1384.35131)]. The authors construct a regularised version of the Heston model and the discrete duality finite volume (DDFV) scheme for this model. The numerical analysis is performed for this scheme and stability estimates on the discrete solution and the discrete gradient are obtained. In addition, the convergence of the DDFV scheme to the weak solution of the regularised Heston model is proven. Numerical experiments are provided in the end of the paper to test the regularisation parameter impact.Netter: probabilistic, stateful network modelshttps://zbmath.org/1472.680122021-11-25T18:46:10.358925Z"Zhang, Han"https://zbmath.org/authors/?q=ai:zhang.han"Zhang, Chi"https://zbmath.org/authors/?q=ai:zhang.chi"Azevedo de Amorim, Arthur"https://zbmath.org/authors/?q=ai:azevedo-de-amorim.arthur"Agarwal, Yuvraj"https://zbmath.org/authors/?q=ai:agarwal.yuvraj"Fredrikson, Matt"https://zbmath.org/authors/?q=ai:fredrikson.matthew"Jia, Limin"https://zbmath.org/authors/?q=ai:jia.liminSummary: We study the problem of using probabilistic network models to formally analyze their quantitative properties, such as the effect of different load-balancing strategies on the long-term traffic on a server farm. Compared to prior work, we explore a different design space in terms of tradeoffs between model expressiveness and analysis scalability, which we realize in a language we call \textit{Netter}. Netter code is compiled to probabilistic automata, undergoing optimization passes to reduce the state space of the generated models, thus helping verification scale. We evaluate Netter on several case studies, including a probabilistic load balancer, a routing scheme reminiscent of MPLS, and a network defense mechanism against link-flooding attacks. Our results show that Netter can analyze quantitative properties of interesting routing schemes that prior work hadn't addressed, for networks of small size (4--9 nodes and a few different types of flows). Moreover, when specialized to simpler, stateless networks, Netter can parallel the performance of previous state-of-the-art tools, scaling up to millions of nodes.
For the entire collection see [Zbl 1471.68017].Optimization of a probabilistic interruption mechanism for cognitive radio networks with prioritized secondary usershttps://zbmath.org/1472.680132021-11-25T18:46:10.358925Z"Zhao, Yuan"https://zbmath.org/authors/?q=ai:zhao.yuan"Yue, Wuyi"https://zbmath.org/authors/?q=ai:yue.wuyiSummary: In this paper, taking the various transmission needs of network users in cognitive radio networks into consideration, we analyze the system performance of cognitive radio networks by considering prioritized secondary users (SUs). The SUs are divided into SUs with higher priority (named SU1) and SUs with lower priority (named SU2). Unlike the preemptive and non-preemptive mechanisms proposed in conventional cognitive radio networks with prioritized SUs, in this paper, we propose a probabilistic interruption mechanism to balance the performance between the two types of SUs. We assume that if an SU1 packet arrives and finds the channel is being occupied by an SU2 packet, this SU1 packet will interrupt the SU2 packet's transmission with a probability (referred to as an interrupting index). In order to adapt to the digital nature of communication networks, based on the system actions of different types of packets, we build a discrete-time Markov chain model to derive the formulas for some important system performance measures. To assess the influence of the interrupting index on the system performance, we demonstrate the numerical results of different performance measures with respect to the interrupting index. Finally, from the perspective of the SU1 packets, we build an optimal function to obtain the optimal interrupting index under different parameter settings.Target counting with Presburger constraints and its application in sensor networkshttps://zbmath.org/1472.680152021-11-25T18:46:10.358925Z"Linker, Sven"https://zbmath.org/authors/?q=ai:linker.sven"Sevegnani, Michele"https://zbmath.org/authors/?q=ai:sevegnani.micheleSummary: One of the applications popularized by the emergence of wireless sensor networks is target counting: the computational task of determining the total number of targets located in an area by aggregating the individual counts of each sensor. The complexity of this task lies in the fact that sensing ranges may overlap, and therefore, targets may be overcounted as, in this setting, they are assumed to be indistinguishable from each other. In the literature, this problem has been proven to be unsolvable, hence the existence of several estimation algorithms. However, the main limitation currently affecting these algorithms is that no \textit{assurance} regarding the precision of a solution can be given. We present a novel algorithm for target counting based on exhaustive enumeration of target distributions using linear Presburger constraints. We improve on current approaches since the estimated counts obtained by our algorithm are by construction guaranteed to be consistent with the counts of each sensor. We further extend our algorithm to allow for weighted topologies and sensing errors for applicability in real-world deployments. We evaluate our approach through an extensive collection of synthetic and real-life configurations.Multi-parametric classification of automaton Markov models based on the sequences they generatehttps://zbmath.org/1472.681072021-11-25T18:46:10.358925Z"Nurutdinova, A. R."https://zbmath.org/authors/?q=ai:nurutdinova.a-r"Shalagin, S. V."https://zbmath.org/authors/?q=ai:shalagin.sergei-viktorovichSummary: This article is devoted to multi-parametric classification of automaton Markov models (AMMs) on the base of output sequences with the use of discriminant analysis. The AMMs under consideration are specified by means of stochastic matrices belonging to subclasses defined a priori. A set of claasification features is introduced to distinguish AMMs specified by matrices from different subclasses. The features are related to the frequency characteristics of sequences generated by AMMs. A method is suggested for determining the minimal length of the sequence need to calculate the features with a required accuracy.Manifold learning with bi-stochastic kernelshttps://zbmath.org/1472.681482021-11-25T18:46:10.358925Z"Marshall, Nicholas F."https://zbmath.org/authors/?q=ai:marshall.nicholas-f"Coifman, Ronald R."https://zbmath.org/authors/?q=ai:coifman.ronald-raphaelSummary: In this paper we answer the following question: what is the infinitesimal generator of the diffusion process defined by a kernel that is normalized such that it is bi-stochastic with respect to a specified measure? More precisely, under the assumption that data is sampled from a Riemannian manifold we determine how the resulting infinitesimal generator depends on the potentially non-uniform distribution of the sample points, and the specified measure for the bi-stochastic normalization. In a special case, we demonstrate a connection to the heat kernel. We consider both the case where only a single data set is given, and the case where a data set and a reference set are given. The spectral theory of the constructed operators is studied, and Nyström extension formulas for the gradients of the eigenfunctions are computed. Applications to discrete point sets and manifold learning are discussed.The secretary problem with a choice functionhttps://zbmath.org/1472.682122021-11-25T18:46:10.358925Z"Kawase, Yasushi"https://zbmath.org/authors/?q=ai:kawase.yasushiSummary: In the classical secretary problem, a decision-maker is willing to hire the best secretary out of \(n\) applicants that arrive in a random order, and the goal is to maximize the probability of choosing the best applicant. In this paper, we introduce the secretary problem with a choice function. The choice function represents the preference of the decision-maker. In this problem, the decision-maker hires some applicants, and the goal is to maximize the probability of choosing the best set of applicants defined by the choice function. We see that the secretary problem with a path-independent choice function generalizes secretary version of the stable matching problem, the maximum weight bipartite matching problem, and the maximum weight base problem in a matroid. When the choice function is path-independent, we provide an algorithm that succeeds with probability at least \(1/e^k\) where \(k\) is the maximum size of the choice, and prove that this is the best possible. Moreover, for the non-path-independent case, we prove that the success probability goes to arbitrary small for any algorithm even if the maximum size of the choice is 2.
For the entire collection see [Zbl 1326.68015].Subpolynomial trace reconstruction for random strings and arbitrary deletion probabilityhttps://zbmath.org/1472.682232021-11-25T18:46:10.358925Z"Holden, Nina"https://zbmath.org/authors/?q=ai:holden.nina"Pemantle, Robin"https://zbmath.org/authors/?q=ai:pemantle.robin"Peres, Yuval"https://zbmath.org/authors/?q=ai:peres.yuval"Zhai, Alex"https://zbmath.org/authors/?q=ai:zhai.alexSummary: The insertion-deletion channel takes as input a bit string \(\mathbf x \in \{0,1\}^n\), and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover \(\mathbf x\) from many independent outputs (called ``traces'') of the insertion-deletion channel applied to \(\mathbf x\). We show that if \(\mathbf x\) is chosen uniformly at random, then \((O(\log^{1/3} n))\) traces suffice to reconstruct \(\mathbf x\) with high probability. For the deletion channel with deletion probability \(q < 1/2\) the earlier upper bound was \(\exp(O(\log^{1/2}n))\). The case of \(q \geq 1/2\) or the case where insertions are allowed has not been previously analyzed, and therefore the earlier upper bound was as for worst-case strings, i.e., \(\exp(O(n^{1/3}))\). We also show that our reconstruction algorithm runs in \(n^{1+o(1)}\) time.
A key ingredient in our proof is a delicate two-step alignment procedure where we estimate the location in each trace corresponding to a given bit of \(\mathbf x\). The alignment is done by viewing the strings as random walks and comparing the increments in the walk associated with the input string and the trace, respectively.A lower bound for the number of elastic collisionshttps://zbmath.org/1472.700382021-11-25T18:46:10.358925Z"Burdzy, Krzysztof"https://zbmath.org/authors/?q=ai:burdzy.krzysztof"Duarte, Mauricio"https://zbmath.org/authors/?q=ai:duarte.mauricio-aIn this paper an improved lower bound for the number \(K(n,d)\) of elastic collisions of \(n\) balls of equal radii and masses in \(d\)-dimensional space is proven. The quantity \(K(n,d)\) is a supremum over all initial conditions, and the collisions counted are of isolated pairs in which the velocities of both balls change by a nonzero amount. To describe the main result a function \(f(n)\) is needed. For a positive integer \(n\geq 3\) and setting \(n_1 = \lfloor n/3\rfloor\) and \(n_2 = n - 2n_1\), the function
\[
f(n) = n_1(n_1+1)n_2 + n_2(n_2 -1)/2 +n_1(n_1 -1)
\]
satisfies \(27 f(n)/ n^3 \to 1\) as \(n\to\infty\), \(f(n) > n^3/27\) for all \(n\geq 3\), \(f(n) > n(n-1)/2\) for \(n\geq 7\), \(f(n) = n(n-1)/2\) for \(n=6\), and \(f(n) < n(n-1)/2\) for \(n\leq 3\leq 5\). The main result is that for all \(n\geq 3\) and all \(d\geq 2\) there holds \(K(n,d) \geq f(n)\).
Since \(f(n) > n^3/27\), this improves the previously known lower bound of \(n^2\) when \(n\geq 3\) and \(d\geq 2\). In comparing the known result \(K(3,2) = 4\) and the elementary lower bound \(K(n,d) \geq n(n-1)/2\) for all \(n\geq 2\) and all \(d\geq 1\), the issue of the sharpness of the elementary lower bound \(K(n,d) \geq n(n-1)/2\) when the number of balls is \(n=4,5,6\), is settled by the next result in the paper: \(K(n,d)\geq K(n,2)\geq 1+n(n-1)/2\) for all \(4\leq n\leq 6\) and all \(d\geq 2\).
The last result of the paper is that the elementary lower bound is not sharp in higher dimension: \(K(d,n) > n(n-1)/2\) for all \(n\geq 3\) and for all \(d\geq 2\).Transmission and reflection at the boundary of a random two-component compositehttps://zbmath.org/1472.741102021-11-25T18:46:10.358925Z"Willis, J. R."https://zbmath.org/authors/?q=ai:willis.john-raymondSummary: A half-space \(x_2 > 0\) is occupied by a two-component statistically uniform random composite with specified volume fractions and two-point correlation. It is bonded to a uniform half-space \(x_2 < 0\) from which a plane wave is incident. The transmitted and reflected mean waves are calculated via a variational formulation that makes optimal use of the given statistical information. The problem requires the specification of the properties of three media: those of the two constituents of the composite and those of the homogeneous half-space. The complexity of the problem is minimized by considering a model acoustic-wave problem in which the three media have the same modulus but different densities. It is formulated as a problem of Wiener-Hopf type which is solved explicitly in the particular case of an exponentially decaying correlation function. Possible generalizations are discussed in a brief concluding section.Study a class of Hilfer fractional stochastic integrodifferential equations with Poisson jumpshttps://zbmath.org/1472.742072021-11-25T18:46:10.358925Z"Balasubramaniam, P."https://zbmath.org/authors/?q=ai:balasubramaniam.pagavathigounder"Saravanakumar, S."https://zbmath.org/authors/?q=ai:saravanakumar.soundararajan"Ratnavelu, K."https://zbmath.org/authors/?q=ai:ratnavelu.kurunathanSummary: In this article, we derive the sufficient conditions for the existence of mild solutions of Hilfer fractional stochastic integrodifferential equations with nonlocal conditions and Poisson jumps in Hilbert spaces. Results will be obtained in the $p$th mean square sense by using the fractional calculus, semigroup theory and stochastic analysis techniques. The article generalizes many of the existing results in the literature in terms of (1) Riemann-Liouville and Caputo derivatives are the special cases. (2) In the sense of $p$th mean square norm. (3) Stochastic integrodifferential with nonlocal conditions and Poisson jumps. A numerical example is provided to validate the obtained theoretical results.Stochastic modelling in fluid dynamics: Itô versus Stratonovichhttps://zbmath.org/1472.760452021-11-25T18:46:10.358925Z"Holm, Darryl D."https://zbmath.org/authors/?q=ai:holm.darryl-dSummary: Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton's principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton's principle requires the Stratonovich process, so we must transform from Itô noise in the \textit{data frame} to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The issue is, `Will non-inertial forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?' This issue will be resolved by elementary considerations.Covariant CP-instruments and their convolution semigroupshttps://zbmath.org/1472.810152021-11-25T18:46:10.358925Z"Heo, Jaeseong"https://zbmath.org/authors/?q=ai:heo.jaeseong"Ji, Un Cig"https://zbmath.org/authors/?q=ai:ji.un-cigSummary: Using probability operators and Fourier transforms of CP-instruments on von Neumann algebras, we give necessary and sufficient conditions for operators to be probability operators associated with covariant CP-instruments or to be Fourier transforms of covariant CP-instruments. We discuss a convolution semigroup of covariant CP-instruments and a semigroup of probability operators associated with CP-instruments on von Neumann algebras.The influence of non-Gaussian noise on weak valueshttps://zbmath.org/1472.810172021-11-25T18:46:10.358925Z"Ma, Fang-Yuan"https://zbmath.org/authors/?q=ai:ma.fang-yuan"Li, Jun-Gang"https://zbmath.org/authors/?q=ai:li.jun-gang"Zou, Jian"https://zbmath.org/authors/?q=ai:zou.jianSummary: The influence of Gaussian noise on weak values is studied. A general expression of weak values is derived, which is applicable to both Markovian and non-Markovian environment. Weak values under random telegraph noise and the colored noise of type \(1/f^a\) are investigated in particular, and the properties of weak values under those non-Gaussian noises are discussed. Furthermore, the threshold time for weak values keeping its characteristic to exceed spectrum range is found, which can reach a large value by the memory effect of non-Markovian environment.Noise-induced multilevel Landau-Zener transitions: density matrix investigationhttps://zbmath.org/1472.810422021-11-25T18:46:10.358925Z"Nyisomeh, I. F."https://zbmath.org/authors/?q=ai:nyisomeh.i-f"Ateuafack, M. E."https://zbmath.org/authors/?q=ai:ateuafack.m-e"Fai, L. C."https://zbmath.org/authors/?q=ai:fai.lukong-corneliusSummary: The generalised multilevel Landau-Zener problem is solved by applying the density matrix technique within the framework of nonstationary perturbation theory. The exact survival probability is achieved as a proof of the Brundobler-Elzer hypothesis [\textit{V. Brundobler}, and \textit{J. Elzer}, ``S-matrix for generalized Landau-Zener problem'', J. Phys. A, Math. Gen. 26, No. 5, 1211--1227 (1993; \url{doi:10.1088/0305-4470/26/5/037})]. The effect of classical Gaussian noise is investigated by averaging the solution over the noise realisation. A generalised formula for slow noise-induced transition probability is obtained and found to agree exactly with all known results. Exact results are reported for the Demkov-Osherov model in the slow and fast noise limits. Thermal transition probabilities are obtained via the activation Arrhenius law and observed to tailor a qubit from thermal decoherence.A new spectral analysis of stationary random Schrödinger operatorshttps://zbmath.org/1472.810922021-11-25T18:46:10.358925Z"Duerinckx, Mitia"https://zbmath.org/authors/?q=ai:duerinckx.mitia"Shirley, Christopher"https://zbmath.org/authors/?q=ai:shirley.christopherThe authors consider random Schrödinger operators of the form
\[
-\Delta + \lambda V_{\omega}
\]
and the associated Schrödinger equation, where \(V_{\omega}\) is a realization of a stationary random potential \(V\). The regime under consideration here is \(0<\lambda \ll 1\). The main goal of the authors is to develop a spectral approach to describe the long time behavior of the system beyond perturbative timescales by using ideas from Malliavin calculus, leading to rigorous Mourre type results. In particular, the authors describe the dynamics by a fibered family of spectral perturbation problems. They then state a number of exact resonance conjectures which would require that Bloch waves exist as resonant modes. An approximate resonance result is obtained and the first spectral proof of the decay of time correlations on the kinetic timescale is also provided.Two faces of Douglas-Kazakov transition: from Yang-Mills theory to random walks and beyondhttps://zbmath.org/1472.811652021-11-25T18:46:10.358925Z"Gorsky, Alexander"https://zbmath.org/authors/?q=ai:gorskii.aleksander-sergeevich"Milekhin, Alexey"https://zbmath.org/authors/?q=ai:milekhin.alexey"Nechaev, Sergei"https://zbmath.org/authors/?q=ai:nechaev.sergei-konstantinovichSummary: Being inspired by the connection between 2D Yang-Mills (YM) theory and (1+1)D ``vicious walks'' (VW), we consider different incarnations of large-\(N\) Douglas-Kazakov (DK) phase transition in gauge field theories and stochastic processes focusing at possible physical interpretations. We generalize the connection between YM and VW, study the influence of initial and final distributions of walkers on the DK phase transition, and describe the effect of the \(\theta\)-term in corresponding stochastic processes. We consider the Jack stochastic process involving Calogero-type interaction between walkers and investigate the dependence of DK transition point on a coupling constant. Relying on the relation between large-\(N 2\) D \(q\)-YM and extremal black hole (BH) with large-\(N\) magnetic charge, we speculate about a physical interpretation of a DK phase transitions in a 4D extremal charged BH.Signal detection using biphotons and potential application in axion-like particle searchhttps://zbmath.org/1472.813372021-11-25T18:46:10.358925Z"Hoang, Le Phuong"https://zbmath.org/authors/?q=ai:hoang.le-phuong"Cao, Xuan Binh"https://zbmath.org/authors/?q=ai:cao.xuan-binhSummary: This paper presents a new optical system for detecting light signals associated with the change in incoming photon number. The system employs quantum correlation of photon pairs created via spontaneous parametric down-conversion (SPDC). The signal, if present, will perturb the flux of the incident photon stream. The perturbed photon stream is first projected through a birefringent crystal where SPDC occurs, converting a single high-energy photon into a pair of low-energy photons. The photons in each pair eventually arrive at separate detectors. By examining the biphoton correlation using the probability distribution of the photons at the detectors, which varies depending on the displacement of the main ``pump'' photon stream and the change in the number of photons, the small optical displacement of the photon stream and its variance can be determined. The change in incident photon number, in other words, the presence of light signal does not influence the average of the measured optical displacement values. Nevertheless, the change in optical displacement measurement variance when the number of incident photons has changed detects the light signal. This optical setup enables the detection of light signals with low noise and remarkably high precision and sensitivity using quantum correlation. The proposed technique has potential application for axion-like particle search in experimental high energy physics.Globe-hoppinghttps://zbmath.org/1472.820032021-11-25T18:46:10.358925Z"Chistikov, Dmitry"https://zbmath.org/authors/?q=ai:chistikov.dmitry-v"Goulko, Olga"https://zbmath.org/authors/?q=ai:goulko.olga"Kent, Adrian"https://zbmath.org/authors/?q=ai:kent.adrian"Paterson, Mike"https://zbmath.org/authors/?q=ai:paterson.mike-sSummary: We consider versions of the grasshopper problem
[the second and third author, Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 473, No. 2207, Article ID 20170494, 19 p. (2017; Zbl 1404.60025)]
on the circle and the sphere, which are relevant to Bell inequalities. For a circle of circumference \(2 \pi \), we show that for unconstrained lawns of any length and arbitrary jump lengths, the supremum of the probability for the grasshopper's jump to stay on the lawn is one. For antipodal lawns, which by definition contain precisely one of each pair of opposite points and have length \(\pi \), we show this is true except when the jump length \(\varphi\) is of the form \(\pi (p/q)\) with \(p, q\) coprime and \(p\) odd. For these jump lengths, we show the optimal probability is \(1 - 1/q\) and construct optimal lawns. For a \textit{pair} of antipodal lawns, we show that the optimal probability of jumping from one onto the other is \(1 - 1/q\) for \(p, q\) coprime, \(p\) odd and \(q\) even, and one in all other cases. For an antipodal lawn on the sphere, it is known
[the third author and \textit{D. Pitalúa-García}, ``Bloch-sphere colorings and Bell inequalities'', Phys. Rev. A 90, Article ID 062124, 13 p. (2014; \url{doi:10.1103/PhysRevA.90.062124})]
that if \(\varphi = \pi /q\), where \(q \in \mathbb{N} \), then the optimal retention probability of \(1 - 1/q\) for the grasshopper's jump is provided by a hemispherical lawn. We show that in all other cases where \(0 < \varphi < \pi /2\), hemispherical lawns are not optimal, disproving the hemispherical colouring maximality hypotheses
[the third author and Pitalúa-García, loc. cit.].
We discuss the implications for Bell experiments and related cryptographic tests.Two-curve Green's function for 2-SLE: the boundary casehttps://zbmath.org/1472.820082021-11-25T18:46:10.358925Z"Zhan, Dapeng"https://zbmath.org/authors/?q=ai:zhan.dapeng.1|zhan.dapengSummary: We prove that for \(\kappa\in (0,8)\), if \((\eta_1,\eta_2)\) is a 2-SLE\(_\kappa\) pair in a simply connected domain \(D\) with an analytic boundary point \(z_0\), then as \(r\to 0^+\), \(P[\mathrm{dist}(z_0,\eta_j)<r,j=1,2]\) converges to a positive number for some \(\alpha>0\), which is called the two-curve Green's function. The exponent \(\alpha\) equals \(\frac{12}{\kappa}-1\) or \(2(\frac{12}{\kappa}-1)\) depending on whether \(z_0\) is one of the endpoints of \(\eta_1\) or \(\eta_2\). We also find the convergence rate and the exact formula for the Green's function up to a multiplicative constant. To derive these results, we construct two-dimensional diffusion processes and use orthogonal polynomials to obtain their transition density.Corrigendum to: ``Exact enumeration of Hamiltonian walks on the \(4\times 4\times 4\) cube and applications to protein folding''https://zbmath.org/1472.820122021-11-25T18:46:10.358925Z"Schram, Raoul D."https://zbmath.org/authors/?q=ai:schram.raoul-d"Schiessel, Helmut"https://zbmath.org/authors/?q=ai:schiessel.helmutFrom the text: The number of Hamiltonian walks on the \(4\times 4\times 4\) cube on pages 12 and 13 of our paper [ibid. 46, No. 48, Article ID 485001, 14 p. (2013; Zbl 1286.82015)] is wrongly stated as 27, 747, 833, 510, 015, 886. The correct number is 27, 746, 717, 207, 772, 000. We have independently verified this number with another method.Glauber dynamics for Ising model on convergent dense graph sequenceshttps://zbmath.org/1472.820182021-11-25T18:46:10.358925Z"Acharyya, Rupam"https://zbmath.org/authors/?q=ai:acharyya.rupam"Stefankovic, Daniel"https://zbmath.org/authors/?q=ai:stefankovic.danielSummary: We study the Glauber dynamics for Ising model on (sequences of) dense graphs. We view the dense graphs through the lens of graphons [\textit{L. Lovász} and \textit{B. Szegedy}, J. Comb. Theory, Ser. B 96, No. 6, 933--957 (2006; Zbl 1113.05092)]. For the ferromagnetic Ising model with inverse temperature \(\beta\) on a convergent sequence of graphs \(\{G_n\}\) with limit graphon \(W\) we show fast mixing of the Glauber dynamics if \(\beta\lambda_1(W)<1\) and slow (torpid) mixing if \(\beta\lambda_1(W)>1\) (where \(\lambda_1(W)\) is the largest eigenvalue of the graphon). We also show that in the case \(\beta\lambda_1(W)=1\) there is insufficient information to determine the mixing time (it can be either fast or slow).
For the entire collection see [Zbl 1372.68012].Phase transition for the interchange and quantum Heisenberg models on the Hamming graphhttps://zbmath.org/1472.820212021-11-25T18:46:10.358925Z"Adamczak, Radosław"https://zbmath.org/authors/?q=ai:adamczak.radoslaw"Kotowski, Michał"https://zbmath.org/authors/?q=ai:kotowski.michal"Miłoś, Piotr"https://zbmath.org/authors/?q=ai:milos.piotrSummary: We study a family of random permutation models on the Hamming graph \(H(2,n)\) (i.e., the 2-fold Cartesian product of complete graphs), containing the interchange process and the cycle-weighted interchange process with parameter \(\theta>0\). This family contains the random walk representation of the quantum Heisenberg ferromagnet. We show that in these models the cycle structure of permutations undergoes a phase transition -- when the number of transpositions defining the permutation is \(\le cn^2\), for small enough \(c>0\), all cycles are microscopic, while for more than \(\geq Cn^2\) transpositions, for large enough \(C>0\), macroscopic cycles emerge with high probability.
We provide bounds on values \(C, c\) depending on the parameter \(\theta\) of the model, in particular for the interchange process we pinpoint exactly the critical time of the phase transition. Our results imply also the existence of a phase transition in the quantum Heisenberg ferromagnet on \(H(2,n)\), namely for low enough temperatures spontaneous magnetization occurs, while it is not the case for high temperatures.
At the core of our approach is a novel application of the cyclic random walk, which might be of independent interest. By analyzing explorations of the cyclic random walk, we show that sufficiently long cycles of a random permutation are uniformly spread on the graph, which makes it possible to compare our models to the mean-field case, i.e., the interchange process on the complete graph, extending the approach used earlier by Schramm.Limit law of a second class particle in TASEP with non-random initial conditionhttps://zbmath.org/1472.820232021-11-25T18:46:10.358925Z"Ferrari, P. L."https://zbmath.org/authors/?q=ai:ferrari.patrik-lino"Ghosal, P."https://zbmath.org/authors/?q=ai:ghosal.promit|ghosal.pratik|ghosal.purnata"Nejjar, P."https://zbmath.org/authors/?q=ai:nejjar.peterIn this paper a totally asymmetric simple exclusion process (TASEP) with non-random initial condition and density \(\lambda\) on \(\mathbb{Z}_-\) and \(\rho\) on \(\mathbb{Z}_+\) as one of the simplest non-reversible interacting particle systems on \(\mathbb{Z}\) lattice is considered. An initial and further particle configurations are assumed and described by the occupation variables \(\{\eta_j\}\). Particles (first-class particles) can jump (they are independent) one step to the right only if their right neighboring site is empty. The particles cannot overtake each other and a labeling to them is associated. The position of particle \(k\) at time \(t\) is denoted by \(x_k(t)\) with the right-to-left ordering. In this paper the second-class particles are considered: when a first-class particle tries to
jump on a site occupied by a second-class particle,
the jump is not suppressed and the two particles interchanges their positions. The applications of second-class particles are very often when the interacting system generates shocks as the discontinuities in the particle density.
The main result of paper is given by Theorem 1.1 which is in the form of the limiting distribution and uses two ingredients: 1) the asymptotic independence of the last passage times from two disjoint initial set of points of a last passage percolation (LPP) model; 2) a tightness-type result on the two LPP problems (by Proposition 3.2 and Corollary 3.4) that extends to general the densities of the Pimentel method.
The paper is divided into two sections where Section 2 shows the connection between TASEP and LPP and the proof of Theorem 1.1, which is mainly based on preliminary results on the control of LPP at different points.Invariant measures for spatial contact model in small dimensionshttps://zbmath.org/1472.820242021-11-25T18:46:10.358925Z"Kondratiev, Yuri"https://zbmath.org/authors/?q=ai:kondratiev.yuri-g"Kutoviy, Oleksandr"https://zbmath.org/authors/?q=ai:kutoviy.oleksandr-v"Pirogov, Sergey"https://zbmath.org/authors/?q=ai:pirogov.sergey-a"Zhizhina, Elena"https://zbmath.org/authors/?q=ai:zhizhina.elena-anatolevnaSummary: We study invariant measures of continuous contact model in small dimensional spaces \((d=1,2)\). We prove that this system has the one-parameter set of invariant measures in the critical regime provided the dispersal kernel has a heavy tail. The convergence to these invariant measures for a broad class of initial states is established.Response theory and phase transitions for the thermodynamic limit of interacting identical systemshttps://zbmath.org/1472.820262021-11-25T18:46:10.358925Z"Lucarini, Valerio"https://zbmath.org/authors/?q=ai:lucarini.valerio"Pavliotis, Grigorios A."https://zbmath.org/authors/?q=ai:pavliotis.grigorios-a"Zagli, Niccolò"https://zbmath.org/authors/?q=ai:zagli.niccoloSummary: We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers-Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker-Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai-Zwanzig model and of the Bonilla-Casado-Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively.Captive diffusions and their applications to order-preserving dynamicshttps://zbmath.org/1472.820272021-11-25T18:46:10.358925Z"Mengütürk, Levent Ali"https://zbmath.org/authors/?q=ai:menguturk.levent-ali"Mengütürk, Murat Cahit"https://zbmath.org/authors/?q=ai:menguturk.murat-cahitSummary: We propose a class of stochastic processes that we call captive diffusions, which evolve within measurable pairs of càdlàg bounded functions that admit bounded right-derivatives at points where they are continuous. In full generality, such processes allow reflection and absorption dynamics at their boundaries -- possibly in a hybrid manner over non-overlapping time periods -- and if they are martingales, continuous boundaries are necessarily monotonic. We employ multi-dimensional captive diffusions equipped with a totally ordered set of boundaries to model random processes that preserve an initially determined rank. We run numerical simulations on several examples governed by different drift and diffusion coefficients. Applications include interacting particle systems, random matrix theory, epidemic modelling and stochastic control.Two-member Markov processes toward an equilibrium from a continuum of initial stateshttps://zbmath.org/1472.820282021-11-25T18:46:10.358925Z"Mok, Jinsik"https://zbmath.org/authors/?q=ai:mok.jinsik"Lee, Hyoung-In"https://zbmath.org/authors/?q=ai:lee.hyoung-inSummary: Dynamics of two-member Markov processes is formulated based on the binomial probability. Sets of initial states are then sought such that the final state reaches an equilibrium. On the two-parameter phase plane, such initial states are found to exhibit diverse geometric configurations depending on the source probability. Those initial-state boundaries undergo phase transitions ranging over pills, stripes, circles, ellipses, lemons, and even fuzzy shapes. These results are quite helpful in understanding several physical phenomena involving photons, electrons, and atoms. For convenience of discussion, deformations of vortices are taken as an example.Mass-based finite volume scheme for aggregation, growth and nucleation population balance equationhttps://zbmath.org/1472.820442021-11-25T18:46:10.358925Z"Singh, Mehakpreet"https://zbmath.org/authors/?q=ai:singh.mehakpreet"Ismail, Hamza Y."https://zbmath.org/authors/?q=ai:ismail.hamza-y"Matsoukas, Themis"https://zbmath.org/authors/?q=ai:matsoukas.themis"Albadarin, Ahmad B."https://zbmath.org/authors/?q=ai:albadarin.ahmad-b"Walker, Gavin"https://zbmath.org/authors/?q=ai:walker.gavinSummary: In this paper, a new mass-based numerical method is developed using the notion of \textit{L. Forestier-Coste} and \textit{S. Mancini} [SIAM J. Sci. Comput. 34, No. 6, B840--B860 (2012; Zbl 1259.82054)] for solving a one-dimensional aggregation population balance equation. The existing scheme requires a large number of grids to predict both moments and number density function accurately, making it computationally very expensive. Therefore, a mass-based finite volume is developed which leads to the accurate prediction of different integral properties of number distribution functions using fewer grids. The new mass-based and existing finite volume schemes are extended to solve simultaneous aggregation-growth and aggregation-nucleation problems. To check the accuracy and efficiency, the mass-based formulation is compared with the existing method for two kinds of benchmark kernels, namely analytically solvable and practical oriented kernels. The comparison reveals that the mass-based method computes both number distribution functions and moments more accurately and efficiently than the existing method.Volumes and random matriceshttps://zbmath.org/1472.830172021-11-25T18:46:10.358925Z"Witten, Edward"https://zbmath.org/authors/?q=ai:witten.edwardSummary: This article is an introduction to newly discovered relations between volumes of moduli spaces of Riemann surfaces or super Riemann surfaces, simple models of gravity or supergravity in two dimensions, and random matrix ensembles. (The article is based on a lecture at the conference on the Mathematics of Gauge Theory and String Theory, University of Auckland, January 2020)Modeling Ocean currents through complex random fields indexed in timehttps://zbmath.org/1472.860042021-11-25T18:46:10.358925Z"Cappello, Claudia"https://zbmath.org/authors/?q=ai:cappello.claudia"De Iaco, Sandra"https://zbmath.org/authors/?q=ai:de-iaco.sandra"Maggio, Sabrina"https://zbmath.org/authors/?q=ai:maggio.sabrina"Posa, Donato"https://zbmath.org/authors/?q=ai:posa.donatoSummary: Surface ocean currents are often of interest in environmental monitoring. These vectorial data can be reasonably treated as a finite realization of a complex-valued random field, where the decomposition in modulus (current speed) and direction (current direction) of the current field is natural. Moreover, when observations are also available for different time points (other than at several locations), it is useful to evaluate the evolution of their complex correlation over time (rather than in space) and the corresponding modeling which is required for estimation purposes. This paper illustrates a first approach where the temporal profile of surface ocean currents is considered. After introducing the fundamental aspects of the complex formalism of a random field indexed in time, a new class of models suitable for including the temporal component is proposed and applied to describe the time-varying complex covariance function of current data. The analysis concerns ocean current observations, taken hourly on 30 April 2016 through high frequency radar systems at some stations located in the Northeastern Caribbean Sea. The selected complex covariance model indexed in time is used for estimation purposes and its reliability is confirmed by a numerical analysis.Blind source separation for compositional time serieshttps://zbmath.org/1472.860272021-11-25T18:46:10.358925Z"Nordhausen, Klaus"https://zbmath.org/authors/?q=ai:nordhausen.klaus"Fischer, Gregor"https://zbmath.org/authors/?q=ai:fischer.gregor"Filzmoser, Peter"https://zbmath.org/authors/?q=ai:filzmoser.peterSummary: Many geological phenomena are regularly measured over time to follow developments and changes. For many of these phenomena, the absolute values are not of interest, but rather the relative information, which means that the data are compositional time series. Thus, the serial nature and the compositional geometry should be considered when analyzing the data. Multivariate time series are already challenging, especially if they are higher dimensional, and latent variable models are a popular way to deal with this kind of data. Blind source separation techniques are well-established latent factor models for time series, with many variants covering quite different time series models. Here, several such methods and their assumptions are reviewed, and it is shown how they can be applied to high-dimensional compositional time series. Also, a novel blind source separation method is suggested which is quite flexible regarding the assumptions of the latent time series. The methodology is illustrated using simulations and in an application to light absorbance data from water samples taken from a small stream in Lower Austria.Optimal road maintenance investment in traffic networks with random demandshttps://zbmath.org/1472.900262021-11-25T18:46:10.358925Z"Passacantando, Mauro"https://zbmath.org/authors/?q=ai:passacantando.mauro"Raciti, Fabio"https://zbmath.org/authors/?q=ai:raciti.fabioSummary: The paper deals with a traffic network with random demands in which some of the roads need maintenance jobs. Due to budget constraints, a central authority has to choose which of them are to be maintained in order to decrease as much as possible the average total travel time spent by all the users, assuming that the network flows are distributed according to the Wardrop equilibrium principle. This optimal road maintenance problem is modeled as an integer nonlinear program, where the objective function evaluation is based on the solution of a stochastic variational inequality. We propose a regularization and approximation procedure for its computation and prove its convergence. Finally, the results of preliminary numerical experiments on some test networks are reported.Endogenous queue number determination in \(G/m/s\) systemshttps://zbmath.org/1472.900282021-11-25T18:46:10.358925Z"Alves, Vasco F."https://zbmath.org/authors/?q=ai:alves.vasco-fSummary: This paper presents a model for the endogenous determination of the number of queues in a \(G/m/s\) system. Customers arriving at a system where s customers are being served play a game, choosing between \(s\) parallel queues or one single queue. Equilibria are obtained for risk-neutral and risk-averse customers. With risk-neutral customers, both a single queue and multiple queues are equilibrium states. When risk-averse customers are considered, there is a unique single queue equilibrium. These results are discussed and suggestions for further research put forth.Asymptotic analysis of reliability of a system with reserve elements and repairing devicehttps://zbmath.org/1472.900292021-11-25T18:46:10.358925Z"Afanas'eva, L. G."https://zbmath.org/authors/?q=ai:afanaseva.larisa-g"Golovastova, E. A."https://zbmath.org/authors/?q=ai:golovastova.e-aSummary: The paper deals with a system consisting of \(n\) identical elements and one repairing device. While one element is working, others stay in reserve. The distribution of working and repairing times of elements are supposed to be exponential. The asymptotic distribution of the system lifetime under the conditions of its high reliability is investigated.Multi-state system analysis with imperfect fault coverage, human error and standby strategieshttps://zbmath.org/1472.900302021-11-25T18:46:10.358925Z"Arora, Ritu"https://zbmath.org/authors/?q=ai:arora.ritu"Tyagi, Vaishali"https://zbmath.org/authors/?q=ai:tyagi.vaishali"Ram, Mangey"https://zbmath.org/authors/?q=ai:ram.mangeySummary: The present work studies the reliability measures of a standby system using coverage factor. The considered system is a combination of main unit and two standby units. Provision of standbys has been taken for smooth functioning of the system. On failure of main unit standby units take the load of main unit and if both standby units failed, system goes to in completely failed state. The failures and repairs of each unit follow exponential and general distribution respectively. The whole system has been analysed under imperfect coverage and human failure. Markov model has been developed to obtain the state transient probabilities. Various reliability measures like availability, MTTF, cost and sensitivity analysis have been evaluated with the help of supplementary variable technique and Laplace transformation. Some graphical illustrations have been taken for better understanding of the model.The effect of network topology on optimal exploration strategies and the evolution of cooperation in a mobile populationhttps://zbmath.org/1472.910082021-11-25T18:46:10.358925Z"Erovenko, Igor V."https://zbmath.org/authors/?q=ai:erovenko.igor-v"Bauer, Johann"https://zbmath.org/authors/?q=ai:bauer.johann"Broom, Mark"https://zbmath.org/authors/?q=ai:broom.mark"Pattni, Karan"https://zbmath.org/authors/?q=ai:pattni.karan"Rychtář, Jan"https://zbmath.org/authors/?q=ai:rychtar.janSummary: We model a mobile population interacting over an underlying spatial structure using a Markov movement model. Interactions take the form of public goods games, and can feature an arbitrary group size. Individuals choose strategically to remain at their current location or to move to a neighbouring location, depending upon their exploration strategy and the current composition of their group. This builds upon previous work where the underlying structure was a complete graph (i.e. there was effectively no structure). Here, we consider alternative network structures and a wider variety of, mainly larger, populations. Previously, we had found when cooperation could evolve, depending upon the values of a range of population parameters. In our current work, we see that the complete graph considered before promotes stability, with populations of cooperators or defectors being relatively hard to replace. By contrast, the star graph promotes instability, and often neither type of population can resist replacement. We discuss potential reasons for this in terms of network topology.Leader-follower consensus on activity-driven networkshttps://zbmath.org/1472.910282021-11-25T18:46:10.358925Z"Hasanyan, Jalil"https://zbmath.org/authors/?q=ai:hasanyan.jalil"Zino, Lorenzo"https://zbmath.org/authors/?q=ai:zino.lorenzo"Lombana, Daniel Alberto Burbano"https://zbmath.org/authors/?q=ai:burbano-lombana.daniel-alberto"Rizzo, Alessandro"https://zbmath.org/authors/?q=ai:rizzo.alessandro"Porfiri, Maurizio"https://zbmath.org/authors/?q=ai:porfiri.maurizioSummary: Social groups such as schools of fish or flocks of birds display collective dynamics that can be modulated by group leaders, which facilitate decision-making toward a consensus state beneficial to the entire group. For instance, leaders could alert the group about attacking predators or the presence of food sources. Motivated by biological insight on social groups, we examine a stochastic leader-follower consensus problem where information sharing among agents is affected by perceptual constraints and each individual has a different tendency to form social connections. Leveraging tools from stochastic stability and eigenvalue perturbation theories, we study the consensus protocol in a mean-square sense, offering necessary-and-sufficient conditions for asymptotic stability and closed-form estimates of the convergence rate. Surprisingly, the prediction of our minimalistic model share similarities with observed traits of animal and human groups. Our analysis anticipates the counterintuitive result that heterogeneity can be beneficial to group decision-making by improving the convergence rate of the consensus protocol. This observation finds support in theoretical and empirical studies on social insects such as spider or honeybee colonies, as well as human teams, where inter-individual variability enhances the group performance.Matrix calculation for ultimate and 1-year risk in the semi-Markov individual loss reserving modelhttps://zbmath.org/1472.910382021-11-25T18:46:10.358925Z"Bettonville, Carole"https://zbmath.org/authors/?q=ai:bettonville.carole"d'Oultremont, Louise"https://zbmath.org/authors/?q=ai:doultremont.louise"Denuit, Michel"https://zbmath.org/authors/?q=ai:denuit.michel-m"Trufin, Julien"https://zbmath.org/authors/?q=ai:trufin.julien"Van Oirbeek, Robin"https://zbmath.org/authors/?q=ai:van-oirbeek.robinA general insurance process is considered. The authors of the paper suppose that between occurrence and closure, a claim may go through several states. A claim is first said to be incurred but not reported (IBNR). During the IBNR period, the insurer is liable for the claim amount but is unaware of the claim's existence. The company is aware of the claim but it may take some time before the first payment is made. The claim is then said to be reported but not paid (RBNP), meaning a reported claim for which no payments have been made yet. Unless the claim closes without payment, it enters the reported but not settled (RBNS) stage where it stays from the first payment until closure, as long as its final cost remains unknown.
The authors of the paper adopt a discrete-time semi-Markov process to describe stages in claim's development. The state space consists in an IBNR state, an RBNP state, a cascade of RBNS states, and two final states corresponding to closure with or without terminal payment. The occupation times in each state are studied with the help of discrete hazard rate. The state-specific discrete hazard rate function corresponds to the probability that a transition takes place given that the claim has stayed in the current state for some time.
The reserving model proposed in this paper extends the approach proposed in
[\textit{K. Antonio} et al., ``A multi-state approach and flexible payment distributions for micro-level reserving in general insurance'', SSRN Preprint, 34 p. (2016; \url{doi:10.2139/ssrn.2777467})].Finite horizon portfolio selection with durable goodshttps://zbmath.org/1472.910432021-11-25T18:46:10.358925Z"Jeon, Junkee"https://zbmath.org/authors/?q=ai:jeon.junkee"Koo, Hyeng Keun"https://zbmath.org/authors/?q=ai:koo.hyeng-keun"Park, Kyunghyun"https://zbmath.org/authors/?q=ai:park.kyunghyunSummary: We study the consumption and portfolio selection problem of a finitely lived agent who derives utility from the stock of durable goods. We show that the agent's effective relative risk aversion implied by the optimal portfolio tends to decline and approaches zero, as the planning horizon approaches, whereas the agent exhibits constant effective relative risk aversion in the infinite horizon model of \textit{A. Hindy} and \textit{C.-F. Huang} [Econometrica 61, No. 1, 85--121 (1993; Zbl 0772.90015)]. The existence of the stock of durable goods acts as buffer stock and induces the highly risk-tolerant attitude. We approach this problem using successive transformations. We transform our problem by applying an isomorphism proposed by
\textit{M. Schroder} and \textit{C. Skiadas} [``An isomorphism between asset pricing models with and without linear habit formation'', Rev. Financ. Stud. 15, No. 4, 1189--1221 (2002; \url{doi:10.1093/rfs/15.4.1189})]
to a singular control problem involving the choice of a monotone increasing consumption process. We next transform the problem into a dual singular control problem using the dual martingale method. We then transform the dual singular control problem into an optimal stopping problem. We analyze the variational inequality arising from the optimal stopping problem and provide an integral representation of optimal strategies.Pricing of fixed-strike arithmetic Asian powered optionshttps://zbmath.org/1472.910462021-11-25T18:46:10.358925Z"Li, Zhen"https://zbmath.org/authors/?q=ai:li.zhen"Xu, Hong-Kun"https://zbmath.org/authors/?q=ai:xu.hong-kunSummary: Fixed-strike arithmetic Asian powered options are introduced in this article. Using the Gamma function, Kummer's confluent hypergeometric function, and parabolic cylinder functions, we derive approximate valuation formulae of these options via the first-order and quadratic Taylor's approximations and the risk-neutral valuation method. Numerical experiments are also included.Unconscious biases in neural populations coding multiple stimulihttps://zbmath.org/1472.920242021-11-25T18:46:10.358925Z"Keemink, Sander W."https://zbmath.org/authors/?q=ai:keemink.sander-w"Tailor, Dharmesh V."https://zbmath.org/authors/?q=ai:tailor.dharmesh-v"van Rossum, Mark C. W."https://zbmath.org/authors/?q=ai:van-rossum.mark-c-wSummary: Throughout the nervous system, information is commonly coded in activity distributed over populations of neurons. In idealized situations where a single, continuous stimulus is encoded in a homogeneous population code, the value of the encoded stimulus can be read out without bias. However, in many situations, multiple stimuli are simultaneously present; for example, multiple motion patterns might overlap. Here we find that when multiple stimuli that overlap in their neural representation are simultaneously encoded in the population, biases in the read-out emerge. Although the bias disappears in the absence of noise, the bias is remarkably persistent at low noise levels. The bias can be reduced by competitive encoding schemes or by employing complex decoders. To study the origin of the bias, we develop a novel general framework based on Gaussian processes that allows an accurate calculation of the estimate distributions of maximum likelihood decoders, and reveals that the distribution of estimates is bimodal for overlapping stimuli. The results have implications for neural coding and behavioral experiments on, for instance, overlapping motion patterns.Nonlinear modeling of neural interaction for spike prediction using the staged point-process modelhttps://zbmath.org/1472.920392021-11-25T18:46:10.358925Z"Qian, Cunle"https://zbmath.org/authors/?q=ai:qian.cunle"Sun, Xuyun"https://zbmath.org/authors/?q=ai:sun.xuyun"Zhang, Shaomin"https://zbmath.org/authors/?q=ai:zhang.shaomin"Xing, Dong"https://zbmath.org/authors/?q=ai:xing.dong"Li, Hongbao"https://zbmath.org/authors/?q=ai:li.hongbao"Zheng, Xiaoxiang"https://zbmath.org/authors/?q=ai:zheng.xiaoxiang"Pan, Gang"https://zbmath.org/authors/?q=ai:pan.gang"Wang, Yiwen"https://zbmath.org/authors/?q=ai:wang.yiwenSummary: Neurons communicate nonlinearly through spike activities. Generalized linear models (GLMs) describe spike activities with a cascade of a linear combination across inputs, a static nonlinear function, and an inhomogeneous Bernoulli or Poisson process, or Cox process if a self-history term is considered. This structure considers the output nonlinearity in spike generation but excludes the nonlinear interaction among input neurons. Recent studies extend GLMs by modeling the interaction among input neurons with a quadratic function, which considers the interaction between every pair of input spikes. However, quadratic effects may not fully capture the nonlinear nature of input interaction. We therefore propose a staged point-process model to describe the nonlinear interaction among inputs using a few hidden units, which follows the idea of artificial neural networks. The output firing probability conditioned on inputs is formed as a cascade of two linear-nonlinear (a linear combination plus a static nonlinear function) stages and an inhomogeneous Bernoulli process. Parameters of this model are estimated by maximizing the log likelihood on output spike trains. Unlike the iterative reweighted least squares algorithm used in GLMs, where the performance is guaranteed by the concave condition, we propose a modified Levenberg-Marquardt (L-M) algorithm, which directly calculates the Hessian matrix of the log likelihood, for the nonlinear optimization in our model. The proposed model is tested on both synthetic data and real spike train data recorded from the dorsal premotor cortex and primary motor cortex of a monkey performing a center-out task. Performances are evaluated by discrete-time rescaled Kolmogorov-Smirnov tests, where our model statistically outperforms a GLM and its quadratic extension, with a higher goodness-of-fit in the prediction results. In addition, the staged point-process model describes nonlinear interaction among input neurons with fewer parameters than quadratic models, and the modified L-M algorithm also demonstrates fast convergence.A mathematical framework for modelling 3D cell motility: applications to glioblastoma cell migrationhttps://zbmath.org/1472.920642021-11-25T18:46:10.358925Z"Scott, M."https://zbmath.org/authors/?q=ai:scott.marian|scott.michael-h|scott.murray|scott.mark|scott.melvin-r|scott.marietta-l-j|scott.m-p-j|scott.mackinley|scott.m-i|scott.mike|scott.marc-a|scott.m-d|scott.matthew|scott.michael-a|scott.michael-b|scott.m-penny|scott.michael-l|scott.michael-j|scott.meckinley"Żychaluk, K."https://zbmath.org/authors/?q=ai:zychaluk.kamila"Bearon, R. N."https://zbmath.org/authors/?q=ai:bearon.rachel-nSummary: The collection of 3D cell tracking data from live images of micro-tissues is a recent innovation made possible due to advances in imaging techniques. As such there is increased interest in studying cell motility in 3D \textit{in vitro} model systems but a lack of rigorous methodology for analysing the resulting data sets. One such instance of the use of these \textit{in vitro} models is in the study of cancerous tumours. Growing multicellular tumour spheroids \textit{in vitro} allows for modelling of the tumour microenvironment and the study of tumour cell behaviours, such as migration, which improves understanding of these cells and in turn could potentially improve cancer treatments. In this paper, we present a workflow for the rigorous analysis of 3D cell tracking data, based on the persistent random walk model, but adaptable to other biologically informed mathematical models. We use statistical measures to assess the fit of the model to the motility data and to estimate model parameters and provide confidence intervals for those parameters, to allow for parametrization of the model taking correlation in the data into account. We use \textit{in silico} simulations to validate the workflow in 3D before testing our method on cell tracking data taken from \textit{in vitro} experiments on glioblastoma tumour cells, a brain cancer with a very poor prognosis. The presented approach is intended to be accessible to both modellers and experimentalists alike in that it provides tools for uncovering features of the data set that may suggest amendments to future experiments or modelling attempts.A simple model for low variability in neural spike trainshttps://zbmath.org/1472.920672021-11-25T18:46:10.358925Z"Ferrari, Ulisse"https://zbmath.org/authors/?q=ai:ferrari.ulisse"Deny, Stéphane"https://zbmath.org/authors/?q=ai:deny.stephane"Marre, Olivier"https://zbmath.org/authors/?q=ai:marre.olivier"Mora, Thierry"https://zbmath.org/authors/?q=ai:mora.thierrySummary: Neural noise sets a limit to information transmission in sensory systems. In several areas, the spiking response (to a repeated stimulus) has shown a higher degree of regularity than predicted by a Poisson process. However, a simple model to explain this low variability is still lacking. Here we introduce a new model, with a correction to Poisson statistics, that can accurately predict the regularity of neural spike trains in response to a repeated stimulus. The model has only two parameters but can reproduce the observed variability in retinal recordings in various conditions. We show analytically why this approximation can work. In a model of the spike-emitting process where a refractory period is assumed, we derive that our simple correction can well approximate the spike train statistics over a broad range of firing rates. Our model can be easily plugged to stimulus processing models, like a linear-nonlinear model or its generalizations, to replace the Poisson spike train hypothesis that is commonly assumed. It estimates the amount of information transmitted much more accurately than Poisson models in retinal recordings. Thanks to its simplicity, this model has the potential to explain low variability in other areas.Autoregressive point processes as latent state-space models: a moment-closure approach to fluctuations and autocorrelationshttps://zbmath.org/1472.920752021-11-25T18:46:10.358925Z"Rule, Michael"https://zbmath.org/authors/?q=ai:rule.michael"Sanguinetti, Guido"https://zbmath.org/authors/?q=ai:sanguinetti.guidoSummary: Modeling and interpreting spike train data is a task of central importance in computational neuroscience, with significant translational implications. Two popular classes of data-driven models for this task are autoregressive point-process generalized linear models (PPGLM) and latent state-space models (SSM) with point-process observations. In this letter, we derive a mathematical connection between these two classes of models. By introducing an auxiliary history process, we represent exactly a PPGLM in terms of a latent, infinite-dimensional dynamical system, which can then be mapped onto an SSM by basis function projections and moment closure. This representation provides a new perspective on widely used methods for modeling spike data and also suggests novel algorithmic approaches to fitting such models. We illustrate our results on a phasic bursting neuron model, showing that our proposed approach provides an accurate and efficient way to capture neural dynamics.Correction to: ``A comparative analysis of noise properties of stochastic binary models for a self-repressing and for an externally regulating gene''https://zbmath.org/1472.921012021-11-25T18:46:10.358925Z"Giovanini, Guilherme"https://zbmath.org/authors/?q=ai:giovanini.guilherme"Sabino, Alan U."https://zbmath.org/authors/?q=ai:sabino.alan-u"Barros, Luciana R. C."https://zbmath.org/authors/?q=ai:barros.luciana-r-c"Ramos, Alexandre F."https://zbmath.org/authors/?q=ai:ramos.alexandre-ferreiraFrom the text: We would like to submit the following corrections to our recently published paper [ibid. 17, No. 5, 5477--5503 (2020; Zbl 1470.92118)] due
to the use of incorrect data files to build two of the graphs contained in the manuscript. We
also corrected some typos in the text that describe the statistical values related to the graphs
mentioned. Additionally, we uploaded a verification code to our Laboratory's GitHub repository
(\url{https://github.com/amphybio/Giovanini2020 ComparativeAnalysis}) to enable the reproduction of our
numerical calculations by the interested readers.Erratum to: ``Extinction and ergodic stationary distribution of a Markovian-switching prey-predator model with additional food for predator''https://zbmath.org/1472.921772021-11-25T18:46:10.358925Z"Guo, Xiaoxia"https://zbmath.org/authors/?q=ai:guo.xiaoxia"Ruan, Dehao"https://zbmath.org/authors/?q=ai:ruan.dehaoErratum to the authors' paper [ibid. 15, Paper No. 46, 18 p. (2020; Zbl 1470.92241)].Deterministic and stochastic mean-field SIRS models on heterogeneous networkshttps://zbmath.org/1472.922002021-11-25T18:46:10.358925Z"Bonaccorsi, Stefano"https://zbmath.org/authors/?q=ai:bonaccorsi.stefano"Turri, Silvia"https://zbmath.org/authors/?q=ai:turri.silviaIn this paper, the authors investigated a model for the spread of an SIRS-type epidemics on a network, both in a deterministic setting and under the presence of a random environment, that enters in the definition of the infection rates of the nodes. The authors then modeled the infection rates in the form of independent stochastic processes. To analyze the problem, the authors applied a mean field approximation that is known as NIMFA model, which allows to get a differential equation for the probability of infection in each node. The authors also obtained a sufficient condition which guarantees the extinction of the epidemics both in the deterministic and in the stochastic setting. The model is novel and the results obtained in this paper could play an important role in network infectious disease modeling.
For the entire collection see [Zbl 1447.00005].\(N\)-intertwined SIS epidemic model with Markovian switchinghttps://zbmath.org/1472.922022021-11-25T18:46:10.358925Z"Cao, Xiaochun"https://zbmath.org/authors/?q=ai:cao.xiaochun"Jin, Zhen"https://zbmath.org/authors/?q=ai:jin.zhenThis paper studies the \(n\)-intertwined SIS epidemic model in continuous-time with Markovian switching, where \(n\) is the number of agents. The model is governed by switched differential equations based on the ordinary differential system for the \(n\)-intertwined model. The paper established the existence and uniqueness of the global positive solution for the system. The stochastic boundedness is studied for the switching system governed by a Markovian chain. Sufficient condition for stochastic extinction is obtained. The paper also estimates the limit of the time average for the behavior of solution. Some numerical examples have also been provided for networks over \(n=1000\) nodes.Dynamical model for social distancing in the U.S. during the COVID-19 epidemichttps://zbmath.org/1472.922032021-11-25T18:46:10.358925Z"Chitanvis, Shirish M."https://zbmath.org/authors/?q=ai:chitanvis.shirish-mSummary: \textbf{Background} Social distancing has led to a ``flattening of the curve'' in many states across the U.S. This is part of a novel, massive, global social experiment which has served to mitigate the COVID-19 pandemic in the absence of a vaccine or effective anti-viral drugs. Hence it is important to be able to forecast hospitalizations reasonably accurately.
\textbf{Methods} We propose on phenomenological grounds a random walk/generalized diffusion equation which incorporates the effect of social distancing to describe the temporal evolution of the probability of having a given number of hospitalizations. The probability density function is log-normal in the number of hospitalizations, which is useful in describing pandemics where the number of hospitalizations is very high.
\textbf{Findings} We used this insight and data to make forecasts for states using Monte Carlo methods. Back testing validates our approach, which yields good results about a week into the future. States are beginning to reopen at the time of submission of this paper and our forecasts indicate possible precursors of increased hospitalizations. However, the trends we forecast for hospitalizations as well as infections thus far show moderate growth. Additionally we studied the reproducibility \textit{Ro} in New York (Italian strain) and California (Wuhan strain). We find that even if there is a difference in the transmission of the two strains, social distancing has been able to control the progression of COVID 19.From ODE to open Markov chains, via SDE: an application to models for infections in individuals and populationshttps://zbmath.org/1472.922112021-11-25T18:46:10.358925Z"Esquível, Manuel L."https://zbmath.org/authors/?q=ai:esquivel.manuel-leote"Patrício, Paula"https://zbmath.org/authors/?q=ai:patricio.paula"Guerreiro, Gracinda R."https://zbmath.org/authors/?q=ai:guerreiro.gracinda-ritaSummary: We present a methodology to connect an ordinary differential equation (ODE) model of interacting entities at the individual level, to an open Markov chain (OMC) model of a population of such individuals, via a stochastic differential equation (SDE) intermediate model. The ODE model here presented is formulated as a dynamic change between two regimes; one regime is of mean reverting type and the other is of inverse logistic type. For the general purpose of defining an OMC model for a population of individuals, we associate an Ito processes, in the form of SDE to ODE system of equations, by means of the addition of Gaussian noise terms which may be thought to model non essential characteristics of the phenomena with small and undifferentiated influences. The next step consists on discretizing the SDE and using the discretized trajectories computed by simulation to define transitions of a finite valued Markov chain; for that, the state space of the Ito processes is partitioned according to some rule. For the example proposed for illustration, the state space of the ODE system referred -- corresponding to a model of a viral infection -- is partitioned into six infection classes determined by some of the critical points of the ODE system; we detail the evolution of some infected population in these infection classes.Classification of asymptotic behavior in a stochastic SEIR epidemic modelhttps://zbmath.org/1472.922212021-11-25T18:46:10.358925Z"Jin, Manli"https://zbmath.org/authors/?q=ai:jin.manli"Lin, Yuguo"https://zbmath.org/authors/?q=ai:lin.yuguoThis paper studies the asymptotic behavior of a stochastic SEIR epidemic model described as \(dS=(\Lambda-\beta S I-\mu S)dt+\sigma_1SdB_1\), \(dE=(\beta SI-(\varepsilon+\mu)E)dt+\sigma_2EdB_2\), \(dI=(\varepsilon E-(\mu+\gamma+\alpha)I)dt+\sigma_3IdB_3\), \(dR=(\gamma I-\mu R)dt+\sigma_4RdB_4\), where \(S, E, I, R\) are the number of susceptible, exposed, infectious and recovered population, respectively. The factors \(\Lambda, \beta, \mu,\varepsilon,\gamma,\alpha\) are positive and \(B_k\) for \(k=1,\cdots, 4\) are independent standard Brownian motions. It is shown \(\lambda\) is a threshold, where \(\lambda<0\) indicates the disease will decay at an exponential rate and \(\lambda>0\) indicates the transition probabilities converge to an invariant measure. It is shown the convergence speed is of a polynomial.Stationary distribution of a stochastic vegetation-water system with reaction-diffusionhttps://zbmath.org/1472.922562021-11-25T18:46:10.358925Z"Pan, Shiliang"https://zbmath.org/authors/?q=ai:pan.shiliang"Zhang, Qimin"https://zbmath.org/authors/?q=ai:zhang.qimin"Meyer-Baese, Anke"https://zbmath.org/authors/?q=ai:meyer-base.ankeSummary: Considering the impact of the mean-reverting Ornstein-Uhlenbeck (O-U) process in an ecosystem, the stable distribution of a stochastic reaction-diffusion vegetation-water system was studied. We proved the existence and uniqueness of the global positive solution of the system by constructing the Lyapunov function. We established the existence and uniqueness of the stable distribution of the solution of the system.Controllability and optimal control for a class of time-delayed fractional stochastic integro-differential systemshttps://zbmath.org/1472.930162021-11-25T18:46:10.358925Z"Sathiyaraj, T."https://zbmath.org/authors/?q=ai:sathiyaraj.t"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrong"Balasubramaniam, P."https://zbmath.org/authors/?q=ai:balasubramaniam.pagavathigounderSummary: In this paper, we study the controllability and optimal control for a class of time-delayed fractional stochastic integro-differential system with Poisson jumps. A set of sufficient conditions is established for complete and approximate controllability by assuming non-Lipschitz conditions and \textit{pth} mean square norm. We also give an existence of optimal control for Bolza problem. Our result is valid for fractional order \(\alpha >\frac{p-1}{p}\), \(p\ge 2.\) Finally, an example is provided to illustrate the efficiency of the obtained theoretical results.Modeling and analysis of network control system based on hierarchical coloured Petri net and Markov chainhttps://zbmath.org/1472.930672021-11-25T18:46:10.358925Z"Li, Jingdong"https://zbmath.org/authors/?q=ai:li.jingdong"Wang, Zhangang"https://zbmath.org/authors/?q=ai:wang.zhangang"Sun, Liankun"https://zbmath.org/authors/?q=ai:sun.liankun"Wang, Wanru"https://zbmath.org/authors/?q=ai:wang.wanruSummary: This paper investigates a modified modeling of networked control systems (NCSs) with programmable logic controller (PLC). First, the controller-to-actuator and sensor-to-controller network-induced delays are investigated by a modeling tactics based on hierarchical coloured Petri net (HCPN) in a structure-conserving way. Comparing with the recent result, the signal transmission delay is set in a random interval instead of a fixed mode; moreover, the data packet drop out and disorder are also taken into consideration. Second, delays captured form CPN tools are analyzed with a strategy based on Baum-Welch algorithm and statistics science. Besides, time delays are modeled as a Markov chain and the transition probabilities is calculated using the consequent from the previous operation. Finally, a comparison verification illustrates the equivalence property between proposed models.On the exponential stability of stochastic perturbed singular systems in mean squarehttps://zbmath.org/1472.931522021-11-25T18:46:10.358925Z"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Ezzine, Faten"https://zbmath.org/authors/?q=ai:ezzine.faten"Hammami, Mohamed Ali"https://zbmath.org/authors/?q=ai:hammami.mohamed-aliSummary: The approach of Lyapunov functions is one of the most efficient ones for the investigation of the stability of stochastic systems, in particular, of singular stochastic systems. The main objective of the paper is the analysis of the stability of stochastic perturbed singular systems by using Lyapunov techniques under the assumption that the initial conditions are consistent. The uniform exponential stability in mean square and the practical uniform exponential stability in mean square of solutions of stochastic perturbed singular systems based on Lyapunov techniques are investigated. Moreover, we study the problem of stability and stabilization of some classes of stochastic singular systems. Finally, an illustrative example is given to illustrate the effectiveness of the proposed results.New Lyapunov conditions of stochastic finite-time stability and instability of nonlinear time-varying SDEshttps://zbmath.org/1472.931662021-11-25T18:46:10.358925Z"Yu, Xin"https://zbmath.org/authors/?q=ai:yu.xin"Yin, Juliang"https://zbmath.org/authors/?q=ai:yin.juliang"Khoo, Suiyang"https://zbmath.org/authors/?q=ai:khoo.suiyangSummary: In this paper, we further consider the problem of stochastic finite-time stability and instability of nonlinear stochastic differential equations (SDEs) with time-varying drift and diffusion terms. The contributions of the paper are highlighted as follows. Some new conditions via multiple Lyapunov functions are given for stochastic finite-time stability of nonlinear time-varying SDEs. Compared with the existing results on stochastic finite-time stability, the constraint of the differential operator \(\mathcal{L}V\) is further relaxed in this paper, which enable us to construct the Lyapunov functions much more easily in applications for nonlinear time-varying SDEs. In addition, a stochastic finite-time instability theorem is proposed, and it shows the necessity of \(lim_{t\to\infty}\int_{t_0}^t c(s) ds = \infty\) in the proposed conditions of stochastic finite-time stability. Some examples are presented to illustrate the new results.Global fixed-time stabilization for switched stochastic nonlinear systems under rational switching powershttps://zbmath.org/1472.931782021-11-25T18:46:10.358925Z"Song, Zhibao"https://zbmath.org/authors/?q=ai:song.zhibao"Li, Ping"https://zbmath.org/authors/?q=ai:li.ping.3|li.ping.5|li.ping|li.ping.1|li.ping.4|li.ping.2"Zhai, Junyong"https://zbmath.org/authors/?q=ai:zhai.junyong"Wang, Zhen"https://zbmath.org/authors/?q=ai:wang.zhen.3"Huang, Xia"https://zbmath.org/authors/?q=ai:huang.xiaSummary: This note studies global fixed-time stabilization for switched stochastic nonlinear systems under rational switching powers. With the aid of stochastic fixed-time stability theorem, a new control approach is presented to ensure that the system state globally reaches zero almost surely in fixed time independent of initial conditions. The developed scheme is used to controller design of the liquid-level system.The existence and uniqueness of viscosity solution to a kind of Hamilton-Jacobi-Bellman equationhttps://zbmath.org/1472.931962021-11-25T18:46:10.358925Z"Hu, Mingshang"https://zbmath.org/authors/?q=ai:hu.mingshang"Ji, Shaolin"https://zbmath.org/authors/?q=ai:ji.shaolin"Xue, Xiaole"https://zbmath.org/authors/?q=ai:xue.xiaoleThe authors study the existence and uniqueness of the viscosity solution to a Hamilton-Jacobi-Bellman (HJB) equation coupled with algebra equations. This kind of equation comes from a stochastic optimal control problem for which the control system is governed by a fully coupled forward-backward stochastic differential equation (FBSDE). By extending Peng's backward semigroup, the authors obtain the dynamic programming principle and prove that the value function is a viscosity solution to the HJB equation mentioned above. By the uniqueness of the solution to FBSDEs, they give a probabilistic approach to study the uniqueness of the solution to this HJB equation. They also show that the value function is the minimum viscosity solution to this HJB equation. When the coefficients are independent of the control variable or the solution is smooth, they prove that the value function is the unique viscosity solution.Convergence of value functions for finite horizon Markov decision processes with constraintshttps://zbmath.org/1472.931972021-11-25T18:46:10.358925Z"Ichihara, Naoyuki"https://zbmath.org/authors/?q=ai:ichihara.naoyukiSummary: This paper is concerned with finite horizon countable state Markov decision processes (MDPs) having an absorbing set as a constraint. Convergence of value iteration is discussed to investigate the asymptotic behavior of value functions as the time horizon tends to infinity. It turns out that the value function exhibits three different limiting behaviors according to the critical value \(\lambda_{\ast}\), the so-called generalized principal eigenvalue, of the associated ergodic problem. Specifically, we prove that (i) if \(\lambda_{\ast}<0\), then the value function converges to a solution to the corresponding stationary equation; (ii) if \(\lambda_{\ast}>0\), then, after a suitable normalization, it approaches a solution to the corresponding ergodic problem; (iii) if \(\lambda_{\ast}=0\), then it diverges to infinity with, at most, a logarithmic order. We employ this convergence result to examine qualitative properties of the optimal Markovian policy for a finite horizon MDP when the time horizon is sufficiently large.A modified MSA for stochastic control problemshttps://zbmath.org/1472.931982021-11-25T18:46:10.358925Z"Kerimkulov, B."https://zbmath.org/authors/?q=ai:kerimkulov.bekzhan"Šiška, D."https://zbmath.org/authors/?q=ai:siska.david"Szpruch, L."https://zbmath.org/authors/?q=ai:szpruch.lukaszSummary: The classical method of successive approximations (MSA) is an iterative method for solving stochastic control problems and is derived from Pontryagin's optimality principle. It is known that the MSA may fail to converge. Using careful estimates for the backward stochastic differential equation (BSDE) this paper suggests a modification to the MSA algorithm. This modified MSA is shown to converge for general stochastic control problems with control in both the drift and diffusion coefficients. Under some additional assumptions the rate of convergence is shown. The results are valid without restrictions on the time horizon of the control problem, in contrast to iterative methods based on the theory of forward-backward stochastic differential equations.Necessary and sufficient conditions of near-optimality in a regime-switching diffusion modelhttps://zbmath.org/1472.931992021-11-25T18:46:10.358925Z"Li, Min"https://zbmath.org/authors/?q=ai:li.min.9|li.min.2|li.min.8|li.min.4|li.min.3|li.min.5|li.min.6|li.min.10|li.min|li.min.1|li.min.7"Wu, Zhen"https://zbmath.org/authors/?q=ai:wu.zhenThe authors consider the stochastic control problem with the system governed by
\[
\begin{cases}dx(t) =b(t,\alpha(t),x(t),u(t))dt+\sigma (t, \alpha(t),x(t),u(t))dB(t),\\ x(s) =y,\end{cases}
\]
where \(\alpha\) is a Markov chain with finite state space, \(u\) the control process, and \(B\) one-dimensional standard Brownian motion. The problem is to minimize a cost function of the form
\[
J(s,y,u(\cdot))=\mathbb{E} \bigg\{\int^T_sf(t, \alpha(t),x(t),u(t))dt+h(x(T))\bigg\}.
\]
Using the variational principle of \textit{I. Ekeland} [J. Math. Anal. Appl. 47, 324--353 (1974; Zbl 0286.49015)], and convex perturbation, necessary conditions for near-optimality control are derived, and it is shown that any near optimal control satisfies the near-maximum principle in some integral sense with order \(\varepsilon^{1/2}\). Under suitable convexity assumptions, sufficient conditions for near-optimality control are obtained. A numerical example is presented along with some simulation results.Optimal linear-quadratic-Gaussian control for discrete-time linear systems with white and time-correlated measurement noiseshttps://zbmath.org/1472.932002021-11-25T18:46:10.358925Z"Liu, Wei"https://zbmath.org/authors/?q=ai:liu.wei|liu.wei.3|liu.wei.8|liu.wei.5|liu.wei.6|liu.wei.7|liu.wei.2|liu.wei.9|liu.wei.1"Shi, Peng"https://zbmath.org/authors/?q=ai:shi.peng|shi.peng.1"Xie, Xiangpeng"https://zbmath.org/authors/?q=ai:xie.xiangpeng"Yue, Dong"https://zbmath.org/authors/?q=ai:yue.dong"Fei, Shumin"https://zbmath.org/authors/?q=ai:fei.shuminSummary: This article is concerned with the optimal linear-quadratic-Gaussian control problem for discrete-time linear systems corrupted by white and time-correlated measurement noises. First, an optimal predictor for the system under consideration is proposed. Then, an optimal controller is designed, which minimizes an expected loss by a control strategy where the control is a function of measurement sequence. The novelty of this article is that a new sequence instead of the original measurement sequence is used to obtain the optimal control scheme where the element in the new sequence is derived from measurement differencing. A verification example is given to illustrate the effectiveness of the developed new design method.A BSDE approach to stochastic linear quadratic control problemhttps://zbmath.org/1472.932042021-11-25T18:46:10.358925Z"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.7|zhang.wei.5|zhang.wei.12|zhang.wei.3|zhang.wei.16|zhang.wei.18|zhang.wei.17|zhang.wei.15|zhang.wei.13|zhang.wei.9|zhang.wei.2|zhang.wei.4|zhang.wei.10|zhang.wei.19|zhang.wei.6|zhang.wei.1"Zhang, Liangquan"https://zbmath.org/authors/?q=ai:zhang.liangquanSummary: In this article, we study a kind of linear quadratic optimal control problem driven by forward-backward stochastic differential equations (FBSDEs in short) with deterministic coefficients. The cost functional is defined by the solution of FBSDEs. By means of the Girsanov transformation, the original issue is turned equivalently into the classical LQ problem. By functional analysis approach, some necessary and sufficient conditions for the existence of optimal controls have been obtained. Moreover, we investigate the relationship between two groups of first-order and second-order adjoint equations. A new stochastic Riccati equation is derived, which leads to the state feedback form of optimal control. By introducing a new Hamiltonian function with an exponential factor, we establish the stochastic maximum principle to deal with the stochastic linear quadratic problem for forward-backward stochastic system with nonconvex control domain using first-order adjoint equation. An illustrative example is given as well.Why class-D audio amplifiers work well: a theoretical explanationhttps://zbmath.org/1472.940212021-11-25T18:46:10.358925Z"Alvarez, Kevin"https://zbmath.org/authors/?q=ai:alvarez.kevin"Urenda, Julio C."https://zbmath.org/authors/?q=ai:urenda.julio-c"Kreinovich, Vladik"https://zbmath.org/authors/?q=ai:kreinovich.vladik-yaSummary: Most current high-quality electronic audio systems use class-D audio amplifiers (D-amps, for short), in which a signal is represented by a sequence of pulses of fixed height, pulses whose duration at any given moment of time linearly depends on the amplitude of the input signal at this moment of time. In this paper, we explain the efficiency of this signal representation by showing that this representation is the least vulnerable to additive noise (that affect measuring the signal itself) and to measurement errors corresponding to measuring time.
For the entire collection see [Zbl 1467.62007].