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.
Publié le : 1985-08-14
Classification:
Diffusion process,
reflecting barrier,
invariant measures,
conformal mapping,
boundary value problem,
60J65,
60K30
@article{1176992906,
author = {Harrison, J. M. and Landau, H. J. and Shepp, L. A.},
title = {The Stationary Distribution of Reflected Brownian Motion in a Planar Region},
journal = {Ann. Probab.},
volume = {13},
number = {4},
year = {1985},
pages = { 744-757},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992906}
}
Harrison, J. M.; Landau, H. J.; Shepp, L. A. The Stationary Distribution of Reflected Brownian Motion in a Planar Region. Ann. Probab., Tome 13 (1985) no. 4, pp. 744-757. http://gdmltest.u-ga.fr/item/1176992906/