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 $\Lambda (D)/(2|D|)$ 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 $\Lambda(D)<0$.