Multidimensional reflected Brownian motions having exponential stationary distributions
type: Item
instance of: scholarly article
title: Multidimensional reflected Brownian motions having exponential stationary distributions
zbMATH Open document ID: 0615.60072
author: J. Michael Harrison
author: R. J. Williams
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: We are concerned with the stationary distribution of reflected Brownian motion (RBM) in a d-dimensional domain G. Such a process behaves like Brownian motion with a constant drift vector \(\mu\) in G and is instantaneously reflected at the boundary, with the direction of reflection given by a non-tangential vector field v on \(\partial G\). We consider first the case where G is smooth and bounded and v varies smoothly over \(\partial G\). It is shown that the RBM has a stationary density of the exponential form \(C(\mu)\exp\{\gamma(\mu)\cdot x\}\) for each \(\mu \in {\mathbb{R}}^ d\) if and only if v satisfies a certain skew symmetry condition. An explicit formula is given for \(\gamma(\mu)\) in terms of v and \(\mu\). Motivated by applications in queueing theory, we next consider the case where G is a convex polyhedral domain and v is constant on each face of the boundary. Postponing for now the treatment of certain foundational questions, we work directly with a basic adjoint relation (BAR) that appears to characterize stationary distributions for a wide class of RBM's in polyhedral domains. This analytic relation is motivated by formal analogy with the smooth case and will be rigorously justified in later work. As in the smooth case, it is found that (BAR) has a solution of exponential form for each \(\mu \in {\mathbb{R}}^ d\) if and only if v satisfies a certain skew symmetry condition. Moreover, under a mild nondegeneracy condition, it is shown that an exponential solution exists for one \(\mu \in {\mathbb{R}}^ d\) if and only if such a solution exists for every \(\mu \in {\mathbb{R}}^ d\).
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 3996799
zbMATH Keywords: stationary distribution of reflected Brownian motion
zbMATH Keywords: stationary density of the exponential form
zbMATH Keywords: applications in queueing theory
zbMATH Keywords: skew symmetry condition
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992259
OpenAlex ID: W2074918857
DOI: 10.1214/AOP/1176992259
description: scientific article; zbMATH DE number 3996799