Exact propagator for generalized Ornstein-Uhlenbeck processes

Abstract

A closed form expression for the propagator is derived, in terms of modified Bessel functions, for the Fokker-Planck equation for a physically important generalization of the Ornstein-Uhlenbeck process where the diffusion constant D(p) is a function of the momentum. The closed form is found for the general case D(p)similar to vertical bar p vertical bar(-alpha) where alpha >= 0 and leads to the standard Gaussian form for alpha=0. The propagator for the specific case D(p)similar to vertical bar p vertical bar(-1) is used to derive analytic expressions for probability distributions and correlation coefficients. An exact expression is found for the constant of proportionality for the anomalous diffusion of the mean-square displacement of a particle at short times.