  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 $\Lambda(D)<0$.
