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March 2013 Shy couplings, $\operatorname{CAT} ({0})$ spaces, and the lion and man
Maury Bramson, Krzysztof Burdzy, Wilfrid Kendall
Ann. Probab. 41(2): 744-784 (March 2013). DOI: 10.1214/11-AOP723

Abstract

Two random processes $X$ and $Y$ on a metric space are said to be $\varepsilon$-shy coupled if there is positive probability of them staying at least a positive distance $\varepsilon$ apart from each other forever. Interest in the literature centres on nonexistence results subject to topological and geometric conditions; motivation arises from the desire to gain a better understanding of probabilistic coupling. Previous nonexistence results for co-adapted shy coupling of reflected Brownian motion required convexity conditions; we remove these conditions by showing the nonexistence of shy co-adapted couplings of reflecting Brownian motion in any bounded $\operatorname{CAT} ({0})$ domain with boundary satisfying uniform exterior sphere and interior cone conditions, for example, simply-connected bounded planar domains with $C^{2}$ boundary.

The proof uses a Cameron–Martin–Girsanov argument, together with a continuity property of the Skorokhod transformation and properties of the intrinsic metric of the domain. To this end, a generalization of Gauss’ lemma is established that shows differentiability of the intrinsic distance function for closures of $\operatorname{CAT} ({0})$ domains with boundaries satisfying uniform exterior sphere and interior cone conditions. By this means, the shy coupling question is converted into a Lion and Man pursuit–evasion problem.

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Maury Bramson. Krzysztof Burdzy. Wilfrid Kendall. "Shy couplings, $\operatorname{CAT} ({0})$ spaces, and the lion and man." Ann. Probab. 41 (2) 744 - 784, March 2013. https://doi.org/10.1214/11-AOP723

Information

Published: March 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1274.60250
MathSciNet: MR3077525
Digital Object Identifier: 10.1214/11-AOP723

Subjects:
Primary: 60J65

Keywords: CAT($\kappa$) , CAT(0) , Co-adapted coupling , coupling , eikonal equation , first geodesic variation , Gauss’ lemma , greedy strategy , Intrinsic metric , Lion and Man problem , Lipschitz domain , Markovian coupling , pursuit–evasion problem , reflected Brownian motion , Reshetnyak majorization , shy coupling , Skorokhod transformation , total curvature , uniform exterior sphere condition , uniform interior cone condition

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 2 • March 2013
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