  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.
