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Short Communications

Diffusion Processes with Reflection and a Third Boundary Value Problem

M. I. Freidlin

Moscow

Abstract: 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).

Received: 31.05.1961

 English version: Theory of Probability and its Applications, 1963, 8:1, 75–83