Open Access
August, 1985 The Stationary Distribution of Reflected Brownian Motion in a Planar Region
J. M. Harrison, H. J. Landau, L. A. Shepp
Ann. Probab. 13(3): 744-757 (August, 1985). DOI: 10.1214/aop/1176992906


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.


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J. M. Harrison. H. J. Landau. L. A. Shepp. "The Stationary Distribution of Reflected Brownian Motion in a Planar Region." Ann. Probab. 13 (3) 744 - 757, August, 1985.


Published: August, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0573.60071
MathSciNet: MR799420
Digital Object Identifier: 10.1214/aop/1176992906

Primary: 60J65
Secondary: 60K30

Keywords: boundary value problem , conformal mapping , diffusion process , Invariant measures , reflecting barrier

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 3 • August, 1985
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