Diffusion Processes with Reflection and a Third Boundary Value Problem

M. I. Freidlin

Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 1, pp. 80-87

Résumé

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).