Abstract: In this paper a kind of numerical method for boundary value problems over bounded or unbounded domains is investigated by probability theory. It's main idea is as follows: Firstly, represent the solution of the boundary value problem as the stochastic representations. Secondly, construct an auxiliary ball and subdivide the boundaries of the domains to make the problem discretized. Finally, by using the strong Markov property and the distributions of the time and place of hitting spheres for Brownian motion with drift, obtain the solutions to the discretized problems.

Key words: Probabilistic numerical method, Boundary value problem, Brownian family with drift.