The factorization and simulation for fundamental solution of Cauchy problem

In this paper, we demonstrate the simulation of fundamental solution for the parabolic equation by the relationship with Ito diffusion. The factorization and Monte Carlo methods of the fundamental solution are considered. With the fact that the fundamental solution can be written as a product of the transition function and the expectation of a bridge path integral, we give an novel and efficient algorithm to simulate the fundamental solution by importance sampling method, especially for dealing with the multi-dimensional case.