Strongly supermedian kernels and Revuz measures
type: Item
instance of: scholarly article
title: Strongly supermedian kernels and Revuz measures
zbMATH Open document ID: 1025.60031
author: Lucian Beznea
author: Nicu Boboc
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: Let \(X\) be a Lusin space, \({\mathcal U}=(U_\alpha)_{\alpha>0}\) a proper sub-Markov resolvent on \(X\) and \(\xi\) an \({\mathcal U}\)-excessive measure. For a strongly supermedian kernel \(V\) on \(X\), define the Revuz measure \(\nu^\xi_V\) by \(\nu^\xi_V(f)=\sup\{\nu(Vf)\); \(\nu\circ U\leq \xi\}\). Further, \(V\) is said regular if, \(Vf \leq s\) on \(\{f>0\}\) for a strongly supermedian function \(s\), then \(Vf \leq s\) everywhere. Analytic characterizations of regular strongly supermedian kernels including Motoo-Mokobodzki property are given. For the excessive measure \(\xi=\rho\circ U+h\) with a harmonic measure \(h\), it is proved that any \(\sigma\)-finite measure charging no set that is both \(\xi\)-polar and \(\rho\)-negligible is a Revuz measure of a regular strongly supermedian kernel. An excessive kernel \(V\) is said natural if, for any Ray open set \(G\) and a function \(f\) vanishing outside of \(G\), \(B^GVf=Vf\), where \(B^GVf\) is the excessive regularization of the réduite of \(Vf\) on \(G\). A characterization of natural excessive kernels by means of the non-charging property of the associated Revuz measure is given. Furthermore, a characterization of the hypothesis (B) of Hunt is given in terms of Revuz measures.
reviewed by: Yoichi Oshima
Mathematics Subject Classification ID: 60J40
Mathematics Subject Classification ID: 31D05
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 31C15
zbMATH DE Number: 1905986
zbMATH Keywords: Revuz measure
zbMATH Keywords: supermedian function
zbMATH Keywords: potential kernel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1008956337
OpenAlex ID: W1977874555
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DOI: 10.1214/AOP/1008956337
description: scientific article; zbMATH DE number 1905986