Monte Carlo method for the Cauchy problem of fractional diffusion equation concerning fractional Laplacian

In this paper, we study a Cauchy problem of fractional diffusion equation concerning fractional Laplacian. We prove the existence and uniqueness of solution to the problem under investigation in Hölder space. Then we apply the Monte Carlo method to solving this Cauchy problem. For the problem with free force term, we derive an unbiased scheme which only produces the statistic error. For the problem with inhomogeneous force term, we establish the simple and high-order jump-adapted schemes by approximating the trajectory of symmetric stable process. The error estimates of these two jump-adapted schemes are given based on the Hölder regularity of the solution. Numerical experiments including even one hundred dimensional case confirm the theoretical analysis and show the numerical efficiency.