Reflected Brownian motion with skew symmetric data in a polyhedral domain

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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 μ 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 in Harrison-Williams [9], 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 [9], a purely analytic problem associated with such an RBM was solved. Here the exponential form solution obtained in [9] 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 in [9] for reflected Brownian motions on smooth approximating domains.