## Illinois Journal of Mathematics

### 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 $\Lambda(D)<0$.

#### Article information

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

Dates
First available in Project Euclid: 12 November 2009

https://projecteuclid.org/euclid.ijm/1258059475

Digital Object Identifier
doi:10.1215/ijm/1258059475

Mathematical Reviews number (MathSciNet)
MR2247829

Zentralblatt MATH identifier
1142.60053

Subjects