The Stationary Distribution of Reflected Brownian Motion in a Planar Region

Suppose given a smooth, compact planar region S and a smooth inward pointing vector field on ∂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.