The stationary distribution of reflected Brownian motion in a planar region. (English) Zbl 0573.60071

Harrison, J. M.; Landau, H. J.; Shepp, L. A.

Ann. Probab. 13, 744-757 (1985).

The present paper deals with the problem of calculating the stationary distribution for a particular type of a two-dimensional diffusion process Z. Its state space is a compact planar region S, and it behaves in the interior of S like a standard Brownian motion. At the boundary Z reflects instantaneously, and the direction of reflection may vary with location.
The original motivation of this work comes from queueing and storage theory. In the first section of the paper the authors give an introductory summary and an explicit statement of their results. In section two the case of smooth state space and smoothly varying direction of reflection is investigated. The main results are the fact that Z is a strong Markov process and the explicit derivation of its stationary probability density function. Section three treats the same topics as section two for the case, that the state space is a polygonal region. Finally in section four some hints to further studies are given.

Reviewer: F.Jondral

MSC:

60J65	Brownian motion
60J60	Diffusion processes
60K30	Applications of queueing theory (congestion, allocation, storage, traffic, etc.)

Keywords:

reflecting barrier; invariant measures; conformal mapping; boundary value problem; stationary distribution; two-dimensional diffusion process; queueing and storage theory