Open Access
January, 1987 Multidimensional Reflected Brownian Motions Having Exponential Stationary Distributions
J. M. Harrison, R. J. Williams
Ann. Probab. 15(1): 115-137 (January, 1987). DOI: 10.1214/aop/1176992259


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 $\nu$ on $\partial G$. We consider first the case where $G$ is smooth and bounded and $\nu$ 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 $\mu \in R^d$ if and only if $\nu$ satisfies a certain skew symmetry condition. An explicit formula is given for $\gamma(\mu)$ in terms of $\nu$ and $\mu$. Motivated by applications in queueing theory, we next consider the case where $G$ is a convex polyhedral domain and $\nu$ 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 $\mu \in R^d$ if and only if $\nu$ satisfies a certain skew symmetry condition. Moreover, under a mild nondegeneracy condition, it is shown that an exponential solution exists for one $\mu \in R^d$ if and only if such a solution exists for every $\mu \in R^d$.


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J. M. Harrison. R. J. Williams. "Multidimensional Reflected Brownian Motions Having Exponential Stationary Distributions." Ann. Probab. 15 (1) 115 - 137, January, 1987.


Published: January, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0615.60072
MathSciNet: MR877593
Digital Object Identifier: 10.1214/aop/1176992259

Primary: 60J65
Secondary: 35J25 , 60J60 , 60K25

Keywords: drift , exponential form , polyhedron , Queueing theory , reflected Brownian motion , skew symmetry , stationary distribution

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 1 • January, 1987
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