Shy couplings, CAT(0) spaces, and the lion and man

TL;DR

This paper proves the nonexistence of shy couplings for reflecting Brownian motion in bounded CAT(0) domains with certain boundary conditions, using advanced probabilistic and geometric techniques, and relates it to pursuit-evasion problems.

Contribution

It removes convexity restrictions from previous nonexistence results, establishing new conditions under which shy couplings cannot exist in CAT(0) domains.

Findings

Nonexistence of shy co-adapted couplings in bounded CAT(0) domains.

Generalization of Gauss' lemma for intrinsic distance differentiability.

Connection of shy coupling problem to Lion and Man pursuit-evasion game.

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 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 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.