A Harnack-type inequality for non-symmetric Markov semigroups
type: Item
instance of: scholarly article
title: A Harnack-type inequality for non-symmetric Markov semigroups
zbMATH Open document ID: 1109.47037
author: Feng-Yu Wang
published in: Journal of Functional Analysis
publication date: 2006-11-15T00:00:00Z
review text: Consider on a locally compact Polish space \(E\) a strong Feller semigroup \(T_t\) which is irreducible, i.e., \(T_t1_G >0\) for any \(t>0\), \(x\in E\) and any open set \(\emptyset \neq G\subset E\). The author shows that \(T_t\) satisfies a Harnack-type inequality of the form \(\sup_E T_tu \leq c_t \inf_E T_tu\) for \(t>0\) and positive measurable functions \(u\) if and only if \(T_t\) admits a unique invariant probability measure \(\pi\) and if \(T_t\) is ultracontractive with respect to the \(L^p\)-scale defined by \(\pi\). For diffusion semigroups on Riemannian manifolds with Ricci curvature bounded from below, the constant \(c_t\) appearing in the Harnack inequality is explicitly calculated and a new condition for ultracontracitvity (hence the Harnack inequality) to hold is given.
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 5072841
zbMATH Keywords: Harnack inequality
zbMATH Keywords: ultracontractivity
zbMATH Keywords: diffusion semigroup
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Ricci curvature
reviewed by: R. L. Schilling
MaRDI profile type: Publication
full work available at URL: j.jfa.2006.06.004
OpenAlex ID: W2136974323
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DOI: 10.1016/J.JFA.2006.06.004
description: scientific article; zbMATH DE number 5072841