Boundary layer analysis on equations of Brownian motion in a force field

The asymptotic analysis of the equations of Brownian motion as treated by Ilin and Khasminskii is extended to include boundaries. By probabilistic methods the boundary layer is analyzed and the boundary condition for the interior solution is derived. It is proven that the interior solution is a uniform approximation inside a compact subset of the region and for times bounded away from zero and infinity. This result is proven for half plane and convex polygonal regions.