Synchronous couplings of reflected Brownian motions in smooth domains

Details Synchronous couplings of reflected Brownian motions in smooth domains Synchronous couplings of reflected Brownian motions in smooth domains

2005-01-27

arxiv.org/abs/math/0501486v1

Mathematics

Probability

Scientific paper

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 $\Lambda(D)<0$.

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