We apply the Monte Carlo method to solving the Dirichlet problem of linear
parabolic equations with fractional Laplacian. This method exploit- s the idea
of weak approximation of related stochastic differential equations driven by
the symmetric stable L\'evy process with jumps. We utilize the jump- adapted
scheme to approximate L\'evy process which gives exact exit time to the
boundary. When the solution has low regularity, we establish a numeri- cal
scheme by removing the small jumps of the L\'evy process and then show the
convergence order. When the solution has higher regularity, we build up a
higher-order numerical scheme by replacing small jumps with a simple process
and then display the higher convergence order. Finally, numerical experiments
including ten- and one hundred-dimensional cases are presented, which confirm
the theoretical estimates and show the numerical efficiency of the proposed
schemes for high dimensional parabolic equations.Comment: 30pages 2 figure