  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.
