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 isometries of nonpositively curved spaces, as well as Diestel–Leader graphs and horocylic products
SHANNON’S THEOREM FOR LOCALLY COMPACT GROUPS / Forghani, B.; Tiozzo, G.. - In: ANNALS OF PROBABILITY. - ISSN 0091-1798. - 50:1(2022), pp. 61-89. [10.1214/21-AOP1529]
SHANNON’S THEOREM FOR LOCALLY COMPACT GROUPS
Tiozzo G.
2022
Abstract
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 isometries of nonpositively curved spaces, as well as Diestel–Leader graphs and horocylic products| File | Dimensione | Formato | |
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