Statistics Papers

Document Type

Journal Article

Date of this Version

1985

Publication Source

The Annals of Probability

Volume

13

Issue

3

Start Page

744

Last Page

757

DOI

10.1214/aop/1176992906

Abstract

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.

Keywords

diffusion process, reflecting barrier, invariant measures, conformal mapping, boundary value problem

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Date Posted: 27 November 2017

This document has been peer reviewed.