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 $\Lambda (D)<0$.