Title
"Brownian motion, Painlevé transcendents, and stochastic integrability: a few examples"
Ivan Dornic (CEA Saclay)
Abstract
We show how the determination of the distribution of a certain imaginary exponential functional of Brownian motion can be recast in terms of the first-passage properties of a particle diffusing in an explicitely time-dependent (sinusoidal) potential. This diffusion turns out to be related to the particular Painlev'e III Sine-Gordon transcendent, in a manner reminiscent of a recent characterization of the celebrated Tracy-Widom distribution. We argue that such a correspondence between non-stationary Schr"odinger (or Fokker-Planck) equations and Painlev'e equations is general. Another salient example is provided by the critical scaling correlations of the two-dimensional Ising model, which are associated to Sinh-Gordon Painlev'e III.
