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An Abramov formula for stationary spaces of discrete groups

Hartman, Yair and Lima, Yuri and Tamuz, Omer (2012) An Abramov formula for stationary spaces of discrete groups. . (Submitted) https://resolver.caltech.edu/CaltechAUTHORS:20161114-064703507

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Abstract

Let (G, μ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A μ-random walk on G, starting from the identity, returns to Γ with probability one. Let θ be the hitting measure, or the distribution of the position in which the random walk first hits Γ. We prove that the Furstenberg entropy of a (G, μ)-stationary space, with respect to the induced action of (Γ, θ), is equal to the Furstenberg entropy with respect to the action of (G, μ), times the index of Γ in G. The index is shown to be equal to the expected return time to Γ. As a corollary, when applied to the Furstenberg-Poisson boundary of (G, μ), we prove that the random walk entropy of (Γ, θ) is equal to the random walk entropy of (G, μ), times the index of Γ in G.


Item Type:Report or Paper (Working Paper)
Related URLs:
URLURL TypeDescription
https://arxiv.org/abs/1204.5414arXivDiscussion Paper
ORCID:
AuthorORCID
Tamuz, Omer0000-0002-0111-0418
Additional Information:Date: April 25, 2012. Submitted on 24 Apr 2012. Yuri Lima is supported by the European Research Council, grant 239885. Omer Tamuz is supported by ISF grant 1300/08, and is a recipient of the Google Europe Fellowship in Social Computing, and this research is supported in part by this Google Fellowship.
Funders:
Funding AgencyGrant Number
European Research Council (ERC)239885
Israel Science Foundation1300/08
GoogleUNSPECIFIED
Subject Keywords:Abramov formula, Furstenberg entropy, random walk, random walk entropy, stationary space
Classification Code:2010 Mathematics Subject Classification. Primary: 37A50, 46L55, 60J50. Secondary: 60J05.
Record Number:CaltechAUTHORS:20161114-064703507
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20161114-064703507
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:71968
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:16 Nov 2016 00:13
Last Modified:03 Oct 2019 16:13

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