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arXiv · 2103.16834

Dynamical behavior of a nonlocal Fokker-Planck equation for a stochastic system with tempered stable noise

Li Lin·Jinqiao Duan·Xiao Wang·Yanjie Zhang
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Abstract

We characterize a stochastic dynamical system with tempered stable noise, by examining its probability density evolution. This probability density function satisfies a nonlocal Fokker-Planck equation. First, we prove a superposition principle that the probability measure-valued solution to this nonlocal Fokker-Planck equation is equivalent to the martingale solution composed with the inverse stochastic flow. This result together with a Schauder estimate leads to the existence and uniqueness of strong solution for the nonlocal Fokker-Planck equation. Second, we devise a convergent finite difference method to simulate the probability density function by solving the nonlocal Fokker-Planck equation. Finally, we apply our aforementioned theoretical and numerical results to a nonlinear filtering system by simulating a nonlocal Zakai equation.

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Li Lin, Jinqiao Duan, Xiao Wang, Yanjie Zhang. 2021-03-31. Dynamical behavior of a nonlocal Fokker-Planck equation for a stochastic system with tempered stable noise. https://doi.org/10.1063/5.0048483

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