Synchronous couplings of reflected Brownian motions in smooth domains

Abstract

For every bounded planar domain D with a smooth boundary, we define a "Lyapunov exponent" [Lambda](D) using a fairly explicit formula. We consider two reflected Brownian motions in D, driven by the same Brownian motion (i.e., a "synchronous coupling"). If [Lambda] (D) > 0 then the distance between the two Brownian particles goes to 0 exponentially fast with rate [Lambda] (D)/(2 [the absolute value of] D) as time goes to infinity. The exponent [Lambda] (D) is strictly positive if the domain has at most one hole. It is an open problem whether there exists a domain with [Lambda](D) < 0.