SHANNON'S THEOREM FOR LOCALLY COMPACT GROUPS

• 成果类型: Article
• 署名作者: Forghani, Behrang; Tiozzo, Giulio
• 署名单位: College of Charleston; University of Toronto
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1529
• 发表日期: 2022
• 页码: 61-89
• 关键词: random-walks; poisson boundary; ergodic theorem; discrete; FORMULA; entropy; PRODUCTS; PROOF

摘要

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.