Reflected Brownian motion with skew symmetric data in a polyhedral domain
type: Item
instance of: scholarly article
title: Reflected Brownian motion with skew symmetric data in a polyhedral domain
zbMATH Open document ID: 0608.60074
DOI: 10.1007/BF00320328
author: R. J. Williams
publication date: 1987-01-01T00:00:00Z
review text: This paper is concerned with the characterization and invariant measures of certain reflected Brownian motions (RBM's) in polyhedral domains. The kind of RBM studied here behaves like d-dimensional Brownian motion with constant drift \(\mu\) in the interior of a simple polyhedron and is instantaneously reflected at the boundary in directions that depend on the face that is hit. Under the assumption that the directions of reflection satisfy a certain skew symmetry condition first introduced by \textit{J. M. Harrison} and the author [Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab., to appear], it is shown that such an RBM can be characterized in terms of a family of submartingales and that it reaches non-smooth parts of the boundary with probability zero. In the quoted paper, a purely analytic problem associated with such an RBM was solved. Here the exponential form solution obtained there is shown to be the density of an invariant measure for the RBM. Furthermore, if the density is integrable over the polyhedral state space, then it yields the unique stationary distribution for the RBM. In the proofs of these results, a key role is played by a dual process for the RBM and by results obtained in the above mentioned paper, for reflected Brownian motions on smooth approximating domains.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60K25
zbMATH DE Number: 3984262
zbMATH Keywords: characterization and invariant measures of certain reflected Brownian motions
zbMATH Keywords: skew symmetry condition
zbMATH Keywords: density of an invariant measure
zbMATH Keywords: unique stationary distribution
MaRDI profile type: Publication
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cites work: Naturality, standardness, and weak duality for Markov processes
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cites work: The diffusion approximation for tandem queues in heavy traffic
cites work: Q3808989
cites work: Reflected Brownian motion on an orthant
cites work: Multidimensional reflected Brownian motions having exponential stationary distributions
cites work: Stochastic differential equations with reflecting boundary conditions
cites work: The adjoint process of a diffusion with reflecting barrier
cites work: Remarks to ``The adjoint process of a diffusion with reflecting barrier''
cites work: Open Queueing Networks in Heavy Traffic
cites work: Q3326564
cites work: Diffusion processes with boundary conditions
cites work: A multidimensional process involving local time
cites work: Stochastic differential equations with reflecting boundary condition in convex regions
cites work: Brownian motion in a wedge with oblique reflection
cites work: Brownian Motion With Polar Drift
cites work: Q5332803
cites work: Q3932100
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 3984262