Fokker-Planck equations for stochastic dynamical systems with symmetric Lévy motions

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• 作者：Ting Gao ^{tinggao0716@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Jinqiao Duan ^{duan@iit.edu" class="auth_mail" title="E-mail the corresponding author} ; Xiaofan Li ; ^{lix@iit.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Non-Gaussian noise ; α-stable symmetric Lé ; vy motion ; Fractional Laplacian operator ; Fokker&ndash ; Planck equation ; Maximum principle ; Toeplitz matrix
• 刊名：Applied Mathematics and Computation
• 出版年：2016
• 出版时间：31 March 2016
• 年：2016
• 卷：278
• 期：Complete
• 页码：1-20
• 全文大小：1297 K

文摘

The Fokker–Planck equations for stochastic dynamical systems, with non-Gaussian α-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 on either a bounded or infinite domain. Under a specified condition, 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.