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&#39; $Λ(D)$ using a fairly explicit formula. We consider two reflected Brownian motions in $D$, driven by the same Brownian motion (i.e., a `synchronous coupling&#39;). 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 $Λ(D)<0$.