Multidimensional Reflected Brownian Motions Having Exponential Stationary Distributions

Abstract

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 $\mu \in R^d$.