Harrison, J. MLandau, H. JShepp, Larry A2023-05-232023-05-2319852016-08-03https://repository.upenn.edu/handle/20.500.14332/47695Suppose 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.diffusion processreflecting barrierinvariant measuresconformal mappingboundary value problemProbabilityArticle