SHANNON'S THEOREM FOR LOCALLY COMPACT GROUPS

We establish the ray and strip approximation criteria for the identification of the Poisson boundary of random walks on locally compact groups. This settles a conjecture from the 1990s by Kaimanovich, who formulated and proved the criterion for discrete groups. The key result is the proof of a version of the Shannon-McMillan-Breiman theorem for locally compact groups. We provide several applications to locally compact groups of isometrics of nonpositively curved spaces, as well as Diestel-Leader graphs and horocylic products.