An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries

TL;DR

This paper establishes an invariance principle for semimartingale reflecting Brownian motions (SRBMs) in domains with piecewise smooth boundaries, providing conditions under which perturbed processes approximate SRBMs and enabling validation of their use in stochastic network models.

Contribution

It proves an invariance principle for SRBMs in complex domains, offering new criteria for their approximation and existence under mild conditions.

Findings

Provides sufficient conditions for processes to approximate SRBMs.

Establishes weak existence of SRBMs in certain domains.

Validates SRBM approximations in convex polyhedrons and bounded domains.

Abstract

Semimartingale reflecting Brownian motions (SRBMs) living in the closures of domains with piecewise smooth boundaries are of interest in applied probability because of their role as heavy traffic approximations for some stochastic networks. In this paper, assuming certain conditions on the domains and directions of reflection, a perturbation result, or invariance principle, for SRBMs is proved. This provides sufficient conditions for a process that satisfies the definition of an SRBM, except for small random perturbations in the defining conditions, to be close in distribution to an SRBM. A crucial ingredient in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. We use the invariance principle to show weak existence of SRBMs under mild conditions. We also use the invariance principle, in conjunction with known uniqueness results for SRBMs, to give some sufficient conditions for validating approximations involving (i) SRBMs in convex polyhedrons with a constant reflection vector field on each face of the polyhedron, and (ii) SRBMs in bounded domains with piecewise smooth boundaries and possibly nonconstant reflection vector fields on the boundary surfaces.