Fokker-Planck Equations for Stochastic Dynamical Systems with Symmetric Lévy Motions

Abstract

The Fokker-Planck equations for stochastic dynamical systems, with non-Gaussian $\alpha -$stable symmetric Lévy motions, have a nonlocal or fractional Laplacian term. This nonlocality is the manifestation of the effect of non-Gaussian fluctuations. Taking advantage of the Toeplitz matrix structure of the time-space discretization, a fast and accurate numerical algorithm is proposed to simulate the nonlocal Fokker-Planck equations, under either absorbing or natural conditions. The scheme is shown to satisfy a discrete maximum principle and to be convergent. It is validated against a known exact solution and the numerical solutions obtained by using other methods. The numerical results for two prototypical stochastic systems, the Ornstein-Uhlenbeck system and the double-well system are shown.

Cited by in corpus (3)

• A regularity result for the nonlocal Fokker-Planck equation with Ornstein-Uhlenbeck drift
• A fast and accurate numerical method for the symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula
• Wellposedness and regularity of steady-state two-sided variable-coefficient conservative space-fractional diffusion equations