Exact propagator for generalized Ornstein-Uhlenbeck processes

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 $D\left(p\right)$ is a function of the momentum. The closed form is found for the general case $D\left(p\right)\sim {\mid p\mid }^{-\alpha }$ where $\alpha ⩾0$ and leads to the standard Gaussian form for $\alpha =0$. The propagator for the specific case $D\left(p\right)\sim {\mid p\mid }^{-1}$ 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.