Abstract In this paper, we study the nonlocal Fokker-Planck equations (FPEs) associated with Levy-driven scalar stochastic dynamical systems. We first derive the Fokker-Planck equation for the case of multiplicative symmetric α-stable noises, by the adjoint operator method. Then we construct a finite difference scheme to simulate the nonlocal FPE on either bounded or infinite domain. It is shown that the semi-discrete scheme satisfies the discrete maximum principle and converges. Some experiments are conducted to validate the numerical method. Finally, we extend the results to the asymmetric case and present an application to the nonlinear filtering problem.

中文翻译：

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具有乘法α稳定噪声的随机动力系统的Fokker-Planck方程的数值分析与应用

摘要 在本文中，我们研究了与 Levy 驱动的标量随机动力系统相关的非局部 Fokker-Planck 方程 (FPE)。我们首先通过伴随算子方法推导出适用于乘法对称 α 稳定噪声情况的 Fokker-Planck 方程。然后我们构造一个有限差分方案来模拟有界域或无限域上的非局部 FPE。结果表明，半离散方案满足离散最大值原理并收敛。进行了一些实验来验证数值方法。最后，我们将结果扩展到非对称情况，并将其应用于非线性滤波问题。