EFFICIENT MONTE CARLO METHOD FOR INTEGRAL FRACTIONAL LAPLACIAN IN MULTIPLE DIMENSIONS

In this paper, we develop a conditional Monte Carlo method for solving PDEs involving an integral fractional Laplacian on any bounded domain in arbitrary dimensions. We first construct the Feynman--Kac representation based on the Green function and Poisson kernel for the fractional Laplacian operator on the unit ball in arbitrary dimensions. Then, a conditional trajectory sampling algorithm is proposed for solving fractional PDEs in the complex domain, inspired by the "walk-on-spheres" algorithm proposed in [A. E. Kyprianou, A. Osojnik, and T. Shardlow, IMA J. Numer. Anal., 38 (2018), pp. 1550-1578]. The proposed method finds it remarkably efficient in solving fractional PDEs: it only needs to evaluate the integrals of expectation form over a sequence of balls maximally inscribed in the domains with the known Green function. Moreover, we prove the proposed method is unbiased and develop bounds on the error and mean stopping time. Finally, ample numerical results are presented to demonstrate the robustness and effectiveness of this approach for fractional PDEs in unit disk and complex domains, and even in ten-dimensional cases.