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.
Funding Statement
G. T. is partially supported by NSERC and the Sloan Foundation.
Acknowledgments
We are greatly indebted to Vadim Kaimanovich for suggesting the problem and providing valuable comments on the first version. We also thank Alex Eskin, Anders Karlsson and Wolfgang Woess for useful comments.
Citation
Behrang Forghani. Giulio Tiozzo. "Shannon’s theorem for locally compact groups." Ann. Probab. 50 (1) 61 - 89, January 2022. https://doi.org/10.1214/21-AOP1529
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