Random walks on discrete groups of polynomial volume growth
type: Item
instance of: scholarly article
title: Random walks on discrete groups of polynomial volume growth
zbMATH Open document ID: 1023.60007
author: Georges Alexopoulos
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: Assume that \(G\) is a finitely generated group of polynomial volume growth. This interesting article studies the asymptotic behavior of the convolution powers \(\mu ^n\) of a probability measure \(\mu\) supported on a finite generating subset of \(G\) and proves many interesting results. We quote the following main results. Upper and lower Gausisan estimates and Berry-Esseen estimates are proved and a central limit theorem is also proved. A parabolic inequality is proved, hence positive \(\mu\)-harmonic functions are constant. A characterization of \(\mu\)-harmonic functions that grow polynomially is also given.
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 1906068
zbMATH Keywords: random walk
zbMATH Keywords: discrete groups of polynomial volume growth
zbMATH Keywords: probability measures
zbMATH Keywords: Gaussian estimates
zbMATH Keywords: Harnack inequality
zbMATH Keywords: harmonic functions
zbMATH Keywords: sub-Laplacian
zbMATH Keywords: heat kernel
reviewed by: C. R. E. Raja
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1023481007
OpenAlex ID: W1968348401
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DOI: 10.1214/AOP/1023481007
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