Sub-Gaussian estimates of heat kernels on infinite graphs
type: Item
instance of: scholarly article
title: Sub-Gaussian estimates of heat kernels on infinite graphs
zbMATH Open document ID: 1010.35016
DOI: 10.1215/S0012-7094-01-10932-0
author: A. A. Grigor'yan
author: Andras Telcs
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: Let \((\Gamma, \mu)\) be a weighted graph that is an infinite connected locally finite graph endowed by the weight \(\mu_{xy}\) for \(x,y\in \Gamma\) and a measure \(\mu(x)=\sum \mu_{xy}\) (where the sum is taken over all \(y\) connected by the wedge with \(x\)) and \(d(x,y)\) be the distance from \(x\) to \(y\). Let \(P(x,y)=\frac{\mu_{xy}}{\mu(x)}\) be a natural Markov operator on the weighted graph, \(P_n(x,y)\) be the \(n\)th convolution power of \(P\) and \(p_n(x,y)=\frac{P_n(x,y)}{\mu(y)}.\) Fixing \(\alpha>\beta>1\) and \(\gamma=\alpha-\beta\) the authors prove that any weighted graph \((\Gamma, \mu)\) with \(P(x,y)\geq p_0>0\) has the regular volume growth \(V(x,R)\simeq R^\alpha,\) \( x\in \Gamma, R\geq 1\) and the estimate \[ \sum_{n=0}^{\infty}p_n(x,y)\simeq d(x,y)^{-\gamma} \quad \text{for } x\neq y \] holds if and only if \[ p_n(x,y)\leq Cn^{\frac{\alpha}{\beta}} \exp\left(- \left(\frac{d(x,y)^\beta}{Cn}\right)^{\frac{1}{\beta-1}}\right) \] and \[ p_n(x,y)+p_{n+1}(x,y)\geq cn^{-\frac{\alpha}{\beta}} \exp\left(- \left(\frac{d(x,y)^\beta}{cn} \right)^{\frac{1}{\beta-1}}\right), \quad n\geq d(x,y) \] for arbitrary \(x,y\in \Gamma\) and a positive integer \(n\).
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 1820844
zbMATH Keywords: locally finite graph
zbMATH Keywords: Markov operator
reviewed by: Yana Belopolskaya
MaRDI profile type: Publication
full work available at URL: s0012-7094-01-10932-0
OpenAlex ID: W1977624157
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description: scientific article; zbMATH DE number 1820844