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The Annals of Probability
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.
diffusion process, reflecting barrier, invariant measures, conformal mapping, boundary value problem
Harrison, J. M., Landau, H. J., & Shepp, L. A. (1985). The Stationary Distribution of Reflected Brownian Motion in a Planar Region. The Annals of Probability, 13 (3), 744-757. http://dx.doi.org/10.1214/aop/1176992906
Date Posted: 27 November 2017
This document has been peer reviewed.