Synchronous couplings of reflected Brownian motions in smooth domains
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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 $Λ(D)<0$.