Diffusion Processes with Reflection and Problems with a Directional Derivative on a Manifold with a Boundary

In this paper a Markov diffusion process with reflection on the boundary of a differentiable manifold is constructed. This construction enables us to investigate the boundary value problem:\[ \sum\limits_{i,j = 1}^n {a_{ij} (x)\frac{{\partial ^2 u}} {{\partial x^i \partial x^j }} + } \sum\limits_{i = 1}^n {b_i (x)} \frac{{\partial u}} {{\partial x^i }} = f(x),\quad \left. {\frac{{\partial u}} {{\partial l}}} \right|_\Gamma = 0, \] using probability methods. Neumann’s problem is a special case of this problem (when l is conformal to the boundary).