Illinois Journal of Mathematics

Synchronous couplings of reflected Brownian motions in smooth domains

Krzysztof Burdzy, Zhen-Qing Chen, and Peter Jones

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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 $\Lambda(D)<0$.

Article information

Illinois J. Math., Volume 50, Number 1-4 (2006), 189-268.

First available in Project Euclid: 12 November 2009

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Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]


Burdzy, Krzysztof; Chen, Zhen-Qing; Jones, Peter. Synchronous couplings of reflected Brownian motions in smooth domains. Illinois J. Math. 50 (2006), no. 1-4, 189--268. doi:10.1215/ijm/1258059475.

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