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

Author

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  • Gao, Ting
  • Duan, Jinqiao
  • Li, Xiaofan

Abstract

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.

Suggested Citation

  • Gao, Ting & Duan, Jinqiao & Li, Xiaofan, 2016. "Fokker–Planck equations for stochastic dynamical systems with symmetric Lévy motions," Applied Mathematics and Computation, Elsevier, vol. 278(C), pages 1-20.
    • Handle: RePEc:eee:apmaco:v:278:y:2016:i:c:p:1-20
    DOI: 10.1016/j.amc.2016.01.010
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    Cited by:

    1. Tesfay, Daniel & Wei, Pingyuan & Zheng, Yayun & Duan, Jinqiao & Kurths, Jürgen, 2020. "Transitions between metastable states in a simplified model for the thermohaline circulation under random fluctuations," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    2. Almaz T. Abebe & Shenglan Yuan & Daniel Tesfay & James Brannan, 2024. "Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise," Mathematics, MDPI, vol. 12(9), pages 1-18, April.
    3. Chen, Xiaoli & Wu, Fengyan & Duan, Jinqiao & Kurths, Jürgen & Li, Xiaofan, 2019. "Most probable dynamics of a genetic regulatory network under stable Lévy noise," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 425-436.
    4. Hao, Mengli & Jia, Wantao & Wang, Liang & Li, Fuxiao, 2022. "Most probable trajectory of a tumor model with immune response subjected to asymmetric Lévy noise," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).

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