Shy Couplings, CAT(0) Spaces, and the Lion and Man

Details Shy Couplings, CAT(0) Spaces, and the Lion and Man Shy Couplings, CAT(0) Spaces, and the Lion and Man

2010-07-19

arxiv.org/abs/1007.3199v3

Mathematics

Probability

v2: 35 pages, 8 figures. This version reformatted from v1 with larger top and bottom margins. v3: 41 pages, 9 figures. Theorem

Scientific paper

Two random processes X and Y on a metric space are said to be epsilon-shy coupled if there is positive probability of them staying at least a positive distance epsilon 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 non-existence results for co-adapted shy coupling of reflected Brownian motion required convexity conditions; we remove these conditions by showing the non-existence of shy co-adapted couplings of reflecting Brownian motion in any bounded CAT(0) domain with boundary satisfying uniform exterior sphere and interior cone conditions, for example, simply-connected bounded planar domains with C2 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 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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