Derivation of Fokker–Planck equations for stochastic systems under excitation of multiplicative non-Gaussian white noise

Abstract

Fokker–Planck equations describe time evolution of probability densities of stochastic dynamical systems and play an important role in quantifying propagation and evolution of uncertainty. Although Fokker–Planck equations can be written explicitly for systems excited by Gaussian white noise, they have remained unknown in general for systems excited by multiplicative non-Gaussian white noise. In this paper, we derive explicit forms of Fokker–Planck equations for one dimensional systems modeled by Marcus stochastic differential equations under multiplicative non-Gaussian white noise. As examples to illustrate the theoretical results, the derived formula is used to obtain Fokker–Planck equations for nonlinear dynamical systems under excitation of (i) α-stable white noise; (ii) combined Gaussian and Poisson white noise, respectively.