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

作者：Su, Bihao; Xu, Chenglong^*; Sheng, Changtao^*

来源：Applied Mathematics Letters, 2023, 141: 108597.

DOI：10.1016/j.aml.2023.108597

• Integral fractional Laplacian
• Transition density
• Poisson kernel
• Monte Carlo method

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.