Weak Convergence of Reflecting Brownian Motions

1. Introduction. We will show that if a sequence of domains D k increases to a domain D then the reflected Brownian motions in D 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. Very recently some progress has been made on the " hot spots " conjecture (Bañuelos and Burdzy [1], Burdzy and Werner [4]), and more papers are being written on the topic. The conjecture was stated in 1974 by J. Rauch and very little was published on the problem since then (see [1] for a review). The conjecture is concerned with the maximum of the second Neumann eigenfunction for the Laplacian in a Euclidean domain. Reflected Brownian motion was used in [1] and [4] to prove the conjecture for some classes of domains and also to give a counterexample. D. Jerison and N. Nadirashvili (private communication) have an argument proving the conjecture for planar convex domains. The question of whether the conjecture holds in planar simply connected domains seems to be the most interesting open problem in the area. A possible way of constructing a counterexample to the " hot spot " conjecture for a class of domains might be first to fix a domain D in the class and consider a sequence of domains D k increasing to D. Then one could consider reflected Brownian motions in D k 's and study their