Open Access
2009 Brownian couplings, convexity, and shy-ness
Wilfrid Kendall
Author Affiliations +
Electron. Commun. Probab. 14: 66-80 (2009). DOI: 10.1214/ECP.v14-1417


Benjamini, Burdzy and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying at least a given positive distance away from each other for all time. Among other results, they showed that no shy couplings could exist for reflected Brownian motions in $C^2$ bounded convex planar domains whose boundaries contain no line segments. Here we use potential-theoretic methods to extend this Benjamini et al.(2007) result (a) to all bounded convex domains (whether planar and smooth or not) whose boundaries contain no line segments, (b) to all bounded convex planar domains regardless of further conditions on the boundary.


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Wilfrid Kendall. "Brownian couplings, convexity, and shy-ness." Electron. Commun. Probab. 14 66 - 80, 2009.


Accepted: 12 February 2009; Published: 2009
First available in Project Euclid: 6 June 2016

MathSciNet: MR2481667
Digital Object Identifier: 10.1214/ECP.v14-1417

Primary: 60J65

Keywords: Brownian motion , coupling

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