Delmotte, Thierry Parabolic Harnack inequality and estimates of Markov chains on graphs. (English) Zbl 0922.60060 Rev. Mat. Iberoam. 15, No. 1, 181-232 (1999). 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 T. Coulhon and 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 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. Reviewer: A.Papadopoulos (Strasbourg) Cited in 2 ReviewsCited in 107 Documents MSC: 60G50 Sums of independent random variables; random walks 31C20 Discrete potential theory 60J45 Probabilistic potential theory Keywords:parabolic Harnack inequality; random walk; Poincaré inequality; reversible Markov chain; Gaussian estimate Citations:Zbl 0792.60063; Zbl 0840.31006 PDF BibTeX XML Cite \textit{T. Delmotte}, Rev. Mat. Iberoam. 15, No. 1, 181--232 (1999; Zbl 0922.60060) Full Text: DOI EuDML OpenURL