Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Levy stable noises

The Fokker-Planck equation has been very useful for studying dynamic behavi or of stochastic differential equations driven by Gaussian noises. However, there are both theoretical and empirical reasons to consider similar equat ions driven by strongly non-Gaussian noises. In particular, they yield stro ngly non-Gaussian anomalous diffusion which seems to be relevant in differe nt domains of Physics. In this paper, we therefore derive a fractional Fokk er-Planck equation for the probability distribution of particles whose moti on is governed by a nonlinear Langevin-type equation, which is driven by a Levy stable noise rather than a Gaussian. We obtain in fact a general resul t for a Markovian forcing. We also discuss the existence and uniqueness of the solution of the fractional Fokker-Planck equation. (C) 2001 American In stitute of Physics.