Synchronous couplings of reflected brownian motions in smooth domains

For every bounded planar domain D with a smooth boundary, we define a "Lyapunov exponent" Λ(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 Λ(D)) > 0 then the distance between the two Brownian particles goes to 0 exponentially fast with rate Λ(D)/(2|D|) as time goes to infinity. The exponent Λ(D) is strictly positive if the domain has at most one hole. It is an open problem whether there exists a domain with Λ(D) < 0.