Rubber bands, pursuit games and shy couplings (Q2874666)

The authors discuss conditions for the existence of capture strategies and geometric consequences of pursuit-evasion games in a disk. The line of reasoning is based on modern metric geometry and the authors define new notions in this area called rubber bands and well-contractible loops, which are used to formulate conditions for a Lion and a Man pursuit-evasion problem (see, e.g. [\textit{R. Isaacs}, Differential games. A mathematical theory with applications to warfare and pursuit, control and optimization. New York, etc.: John Wiley and Sons, Inc. (1965; Zbl 0125.38001)]) under which the Man can either evade the Lion or the Lion can capture the Man. The former conditions are related to the existence of stable rubber bands and the latter to well-contractibility of all loops. Then the same machinery is used to find conditions for the existence of shy couplings (see [\textit{I. Benjamini} et al., Probab. Theory Relat. Fields 137, No. 3--4, 345--377 (2007; Zbl 1113.60076)]) for reflected Brownian motions. The theory is illustrated by a number of examples.