2005/01/27 by Krzysztof Burdzy, Zhen-Qing Chen, Burdzy, Krzysztof +4
Economics, Econometrics and Finance · Mathematics · #60J55 #60J65 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60J55 #msc:60J65
paper · pdf · doi:10.48550/arxiv.math/0501486
arxiv created 2005/01/27 · openalex publication_date 2005/01/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.