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) | ||||||||
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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. | ||||||||
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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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