A new ?walk on spheres? type method for fractional diffusion equation in high dimensions based on the Feynman-Kac formulas

摘要

In this paper, we introduce an efficient stochastic method for solving space -fractional diffusion equations in high dimensions based on the Feynman-Kac formula. The key idea is to approximate the trajectory of the process by using a series of balls. As an extension of our first work Sheng et al. (2022) for the Poisson equation, the new algorithm finds remarkably efficient in solving time-dependent linear problems with integral fractional Laplacian on the bounded and unbounded domains. We present some numerical examples to validate the robustness and efficiency of our methods.

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