Parabolic Harnack inequality and estimates of Markov chains on graphs
type: Item
instance of: scholarly article
title: Parabolic Harnack inequality and estimates of Markov chains on graphs
zbMATH Open document ID: 0922.60060
DOI: 10.4171/RMI/254
author: Thierry Delmotte
published in: Revista Matemática Iberoamericana
publication date: 1999-10-03T00:00:00Z
full work available at URL: https://eudml.org/doc/39568
review text: Let \(\Gamma\) be a graph and let \(p_n(x,y)\) be the kernel of the standard random walk on \(\Gamma\). The author is interested in finding conditions under which one has the following Gaussian estimate: \[ {c\over V(x,\sqrt{n})}e^{-Cd(x,y)^{2}/n}\leq p_n(x,y)\leq {C\over V(x,\sqrt{n})}e^{-cd(x,y)^{2}/n} \] for some constants \(c\) and \(C\), where \(V(x,n)\) is the cardinality of the ball of center \(x\) and radius \(n\), with the assumptions that \(d(x,y)\leq n\) and that all the vertices of \(\Gamma\) are loops. The author proves that the inequalities hold for graphs of polynomial growth under an isoperimetric assumption such as Poincaré inequality. This proves a conjecture made by \textit{T. Coulhon} and \textit{L. Saloff-Coste} [Probab. Theory Relat. Fields 97, No. 3, 423-431 (1993; Zbl 0792.60063)]. The author proves in fact a characterization of the parabolic Harnack inequality. The result is a discrete counterpart of a result of \textit{L. Saloff-Coste} [Potential Anal. 4, No. 4, 429-467 (1995; Zbl 0840.31006)]. The author gives, as an application of the Harnack inequality, a new proof of the theorem of J. Nash on the Hölder regularity for solutions of the elliptic/parabolic equation.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 31C20
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 1275491
zbMATH Keywords: parabolic Harnack inequality
zbMATH Keywords: random walk
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: reversible Markov chain
zbMATH Keywords: Gaussian estimate
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1275491