Fokker-Planck equation driven by asymmetric Lévy motion
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Non-Gaussian Lévy noises are present in many models for understanding underlining principles of physics, finance, biology, and more. In this work, we consider the Fokker-Planck equation (FPE) due to one-dimensional asymmetric Lévy motion, which is a non-local partial differential equation. We present an accurate numerical quadrature for the singular integrals in the non-local FPE and develop a fast summation method to reduce the order of the complexity from O(J2) to \(O(J\log J)\) in one time step, where J is the number of unknowns. We also provide conditions under which the numerical schemes satisfy maximum principle. Our numerical method is validated by comparing with exact solutions for special cases. We also discuss the properties of the probability density functions and the effects of various factors on the solutions, including the stability index, the skewness parameter, the drift term, the Gaussian and non-Gaussian noises, and the domain size.
KeywordsFokker-Planck equations Non-Gaussian noises Asymmetric α-stable Lévy motion Non-local partial differential equation Fast algorithm
Mathematics Subject Classification (2010)65M06 35Q84
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The research is partially supported by the grants China Scholarship Council no. 201306160071 (X.W.), NSF-DMS no. 1620449 (J.D. and X.L.), and NNSFs of China nos. 11531006 and 11771449 (J.D.).
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