On the closable parts of pre-Dirichlet forms and the fine supports of underlying measures
type: Item
instance of: scholarly article
title: On the closable parts of pre-Dirichlet forms and the fine supports of underlying measures
zbMATH Open document ID: 0756.60071
author: Setsuo Taniguchi
author: Masatoshi Fukushima
author: Ken-Iti Sato
published in: Osaka Journal of Mathematics
publication date: 1992-09-27T00:00:00Z
review text: Let \(({\mathcal E},{\mathcal F})\) be an either irreducible or transient regular Dirichlet form on \(L^ 2(X;m)\) where \(X\) is a locally compact separable metric space, \(m\in{\mathcal M}\) with \(\text{supp}[m]=X\), and \({\mathcal M}:=\{\nu\mid \nu\) positive Radon measure on \(X\}\). Let \({\mathcal C}\subset{\mathcal F}\) be a dense subalgebra of continuous functions with compact support on \(X\) such that \(({\mathcal C}.1)\) For every \(K,G\subset X\) with \(K\subset G\), \(K\) compact, \(G\) relatively compact and open, there exists \(w\in{\mathcal C}\) such that \(0\leq w\leq 1\), \(w=1\) on \(K\) and \(w=0\) on \(X\setminus G\). \(({\mathcal C}.2)\) For every \(\varepsilon>0\) there exists \(\beta_ \varepsilon:\mathbb{R}\to[-\varepsilon,1+\varepsilon]\) such that \(\beta_ \varepsilon(t)=t\) for all \(t\in[0,1]\) and \(0\leq\beta_ \varepsilon(t)- \beta_ \varepsilon(s)\leq t-s\) for \(t>s\) and \(\beta_ \varepsilon(f)\in{\mathcal C}\) whenever \(f\in{\mathcal C}\). Let \(Cap\) denote the 1-capacity associated with \(({\mathcal E},{\mathcal F})\). Let \({\mathcal M}_ 0:=\{\nu\mid \nu\) charges no Cap-zero set\(\}\), \({\mathcal M}_{00}:=\{\nu\in{\mathcal M}_ 0\mid Cap(X\setminus\tilde S_ \nu)=0\}\), where \(\tilde S_ \nu\) denotes the support of the positive continuous additive functional (for the underlying Hunt process) associated with \(\nu\in{\mathcal M_ 0}\). The first result of this very interesting paper is that each \(\mu\in{\mathcal M}\) has a unique decomposition \[ \mu=\mu_ 0+\mu_ 1,\quad \mu_ 0\in{\mathcal M}_ 0, \quad \mu_ 1=1_ N\cdot\mu\text{ with }Cap(N)=0. \] The main result states then that for \(\mu\in{\mathcal M}\) with \(\text{supp}[\mu]=X\), \[ ({\mathcal E},{\mathcal C})\text{ is closable on }L^ 2(X;\mu)\text{ if and only if }\mu_ 0\in{\mathcal M}_{00}.\tag{1} \] The techniques of the paper combine in a very nice way analytic and probabilistic methods. In particular, the existence of the closable part of \(({\mathcal E},{\mathcal C})\) …
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 60J40
zbMATH DE Number: 59705
zbMATH Keywords: regular Dirichlet form
zbMATH Keywords: additive functional
zbMATH Keywords: Hunt process
zbMATH Keywords: existence of a largest closable form
zbMATH Keywords: regularization techniques
zbMATH Keywords: Dirichlet forms
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 59705