Existence of positive harmonic functions on groups and on covering manifolds
type: Item
instance of: scholarly article
title: Existence of positive harmonic functions on groups and on covering manifolds
zbMATH Open document ID: 0833.22006
author: Philippe Bougerol
author: Laure Elie
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1995-04-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77509
review text: The authors continue the study of bounded or positive harmonic functions on locally compact groups and on Riemannian manifolds. Main results: (1) Let \(G\) be a compactly generated group for which there is a continuous homomorphism \(f\) from \(G\) into an almost connected group \(L\) (i.e., \(L\) has a co-compact connected subgroup) such that the closure of \(f(G)\) has exponential growth. Then any centered adapted probability measure on \(G\) with a third moment has non-constant continuous positive harmonic functions. (2) Let \(M\) be a regular covering of a compact manifold such that the deck transformation group is a closed subgroup of an almost connected group. Then there exist non-constant positive harmonic functions on \(M\) if and only if \(M\) is of exponential growth. In particular, a centered probability measure on a linear group of exponential growth has non-constant positive harmonic functions. The same conclusion holds for the Laplace-Beltrami operator on a co-compact Riemannian covering with such a covering group.
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 60J50
zbMATH DE Number: 740318
zbMATH Keywords: random walks
zbMATH Keywords: bounded harmonic functions
zbMATH Keywords: positive harmonic functions
zbMATH Keywords: locally compact groups
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: probability measure
zbMATH Keywords: regular covering
zbMATH Keywords: compact manifold
zbMATH Keywords: deck transformation group
zbMATH Keywords: Laplace- Beltrami operator
zbMATH Keywords: co-compact Riemannian covering
reviewed by: Bohumil František Šmarda
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 740318