The Stationary Distribution of Reflected Brownian Motion in a Planar Region

Abstract

Suppose given a smooth, compact planar region $S$ and a smooth inward pointing vector field on $\partial S$. It is known that there is a diffusion process $Z$ which behaves like standard Brownian motion inside $S$ and reflects instantaneously at the boundary in the direction specified by the vector field. It is also known $Z$ has a stationary distribution $p$. We find a simple, general explicit formula for $p$ in terms of the conformal map of $S$ onto the upper half plane. We also show that this formula remains valid when $S$ is a bounded polygon and the vector field is constant on each side. This polygonal case arises as the heavy traffic diffusion approximation for certain two-dimensional queueing and storage processes.