Rubber bands, pursuit games and shy couplings

Abstract

In this paper, we consider pursuit-evasion and probabilistic consequences of some geometric notions for bounded and suitably regular domains in Euclidean space that are CAT(?) for some. These geometric notions are useful for analysing the related problems of (a) existence/nonexistence of successful evasion strategies for the Man in Lion and Man problems, and (b) existence/non-existence of shy couplings for reflected Brownian motions. They involve properties of rubber bands and the extent to which a loop in the domain in question can be deformed to a point without, in between, increasing its loop length. The existence of a stable rubber band will imply the existence of a successful evasion strategy but, if all loops in the domain are wellcontractible, then no successful evasion strategy will exist and there can be no co-adapted shy coupling. For example, there can be no shy couplings in bounded and suitably regular starshaped domains and so, in this setting, any two reflected Brownian motions must almost surely make arbitrarily close encounters as t?8. © 2014 London Mathematical Society.