On the stationary measure of a reflected Brownian motion in a wedge:
some explicit results
(joint work with J. Moriarty, University of Manchester)
Multi-dimensional reflected Brownian motion is a Markov process which
plays an important role in applications. It is particularly widely used
to approximate the behavior of heavily loaded queueing networks. Of
special interest is the long-term behavior of the process, i.e., its
stationary measure.
Although the stationary distribution of one-dimensional reflected
Brownian motion with drift is exponential, this is generally not true
in a multidimensional setting. M. Harrison and R. Williams have
characterized the class of reflected Brownian motions with exponential
densities in terms of a so-called skew symmetry condition on the
reflection directions.
However, not much is known on the stationary measure (or its density)
when the skew symmetry condition fails to hold. For special
multidimensional reflected Brownian motions, an intriguingly simple
formula arises from connections with reflection groups. In this
formula, the stationary density is represented as a finite sum of
exponential terms.
This talk reports an attempt to marry this formula with the literature on two-dimensional reflected Brownian motion in a wedge.