Shy couplings, $\operatorname{CAT} ({0})$ spaces, and the lion and man

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.