Local solution for a class of mixed boundary value problems

A local method is developed for solving locally partial differential equations with mixed boundary conditions. The method is based on a heuristic idea, properties of diffusion processes, stopping times and the It (o) over cap formula for semimartingales. According to the heuristic idea, the diffusion process used for solving locally a partial differential with mixed boundary conditions is stopped when it reaches a Neumann boundary and then restarted inside the domain of definition of this equation at a point depending on the Neumann conditions. The proposed method is illustrated and its accuracy assessed by two simple numerical examples solving locally mixed boundary value problems in one and two space dimensions.