Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedrons

We consider the problem of existence and uniqueness of semimartingale reflecting Brownian motions (SRBM's) in convex polyhedrons. Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the polyhedron the process behaves like a Brownian motion with a constant drift and covariance matrix, and at each of the $(d-1)$-dimensional faces that form the boundary of the polyhedron, the bounded variation part of the process increases in a given direction (constant for any particular face), so as to confine the process to the polyhedron. For historical reasons, this “pushing” at the boundary is called instantaneous reflection. For simple convex polyhedrons, we give a necessary and sufficient condition on the geometric data for the existence and uniqueness of an SRBM. For nonsimple convex polyhedrons, our condition is shown to be sufficient. It is an open question as to whether our condition is also necessary in the nonsimple case. From the uniqueness, it follows that an SRBM defines a strong Markov process. Our results are applicable to the study of diffusions arising as heavy traffic limits of multiclass queueing networks and in particular, the nonsimple case is applicable to multiclass fork and join networks. Our proof of weak existence uses a patchwork martingale problem introduced by T. G. Kurtz, whereas uniqueness hinges on an ergodic argument similar to that used by L. M. Taylor and R. J. Williams to prove uniqueness for SRBM's in an orthant.