Weak convergence of reflecting Brownian motions

We will show that if a sequence of domains D [subscript] k increases to a domain D then the reflected Brownian motions in D [subscript] k's converge to the reflected Brownian motion in D, under mild technical assumptions. Our theorem follows easily from known results and is perhaps known as a "folk law" among the specialists but it does not seem to be recorded anywhere in an explicit form. The purpose of this note is to fill this gap. As the theorem itself is not hard to prove, we will start with some remarks explaining the significance of the result in the context of a currently active research area.Burdzy's research supported in part by NSF grant DMS-9700721. Chen's research supported in part by NSA grant MDA904-98-1-0044