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

Abstract

The Fokker–Planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by Gaussian noises. However, there are both theoretical and empirical reasons to consider similar equations driven by strongly non-Gaussian noises. In particular, they yield strongly non-Gaussian anomalous diffusion which seems to be relevant in different domains of Physics. In this paper, we therefore derive a fractional Fokker–Planck equation for the probability distribution of particles whose motion is governed by a nonlinear Langevin-type equation, which is driven by a Lévy stable noise rather than a Gaussian. We obtain in fact a general result for a Markovian forcing. We also discuss the existence and uniqueness of the solution of the fractional Fokker–Planck equation.