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

TL;DR

This paper derives explicit Fokker-Planck equations for one-dimensional nonlinear stochastic systems driven by multiplicative non-Gaussian white noise, expanding the analytical tools available for uncertainty propagation in such complex systems.

Contribution

The paper introduces explicit Fokker-Planck equations for systems modeled by Marcus stochastic differential equations under non-Gaussian noise, a significant extension beyond Gaussian cases.

Findings

Derived formulas for systems with alpha-stable white noise.

Fokker-Planck equations for systems with combined Gaussian and Poisson noise.

Illustrated theoretical results with practical examples.

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 nonlinear dynamical systems excited by Gaussian white noise, they are not available in general for nonlinear dynamical systems excited by multiplicative non-Gaussian white noise. Marcus stochastic differential equations are often appropriate models in engineering and physics for stochastic dynamical systems excited by 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-Plank equations for nonlinear dynamical systems under excitation of (i) $\alpha$-stable white noise; (ii) combined Gaussian and Poisson white noise, respectively.