Shy couplings, \(\mathrm{CAT}(0)\) spaces, and the lion and man (Q1951689)

This paper deals with so-called shy couplings: the two random process \(X\) and \(Y\) on metric space are said to be \(\epsilon\)-shy coupled if there is an \(\epsilon>0\) such that \(P[\mathrm{dist}(X(t),Y(t))\geq\epsilon\text{ for all }t]>0\). Interest in the existence or nonexistence of such couplings arises from the study of couplings of reflected Brownian motion, which occur in various contexts. Previous nonexistence results for co-adapted shy coupling of reflected Brownian motion required convexity conditions; the authors 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.