Time changes in Dirichlet space theory
type: Item
instance of: scholarly article
title: Time changes in Dirichlet space theory
zbMATH Open document ID: 0764.31005
author: Kazuhiro Kuwae
author: Shintaro Nakao
published in: Osaka Journal of Mathematics
publication date: 1993-01-16T00:00:00Z
review text: Let \(X\) be an irreducible \(C_ 0\)-regular \(m\)-symmetric Hunt process on a locally compact separable state space, and \(({\mathcal E},{\mathcal F})\) denote the associated Dirichlet space on \(L^ 2(m)\). Let \(A=(A_ t)_{t\geq0}\) be a positive continuous additive functional of \(X\) with Revuz measure \(\mu\) and fine support \(\tilde Y\). It is well-known that the time-changed process \(\tilde X_ t:=X_{A^{-1}(t)}\) is a \(\mu\)-symmetric strong Markov process with natural state space \(\tilde Y\). Let \(Y\) denote the topological support of \(\mu\). Then \(\tilde Y\backslash Y\) is of zero capacity and \(\mu(Y\backslash\tilde Y)=0\), so \(\tilde X\) may be viewed as a \(\mu\)-symmetric process on \(Y\). The authors show that after modification on an exceptional set, \(\tilde X\) becomes a Hunt process. Moreover, the associated Dirichlet space is \(C_ 0\)-regular if \(Y\) is taken as the base space, and the authors identify this Dirichlet space explicitly. These results extend and complete work of Fukushima, Silverstein, Oshima, and the reviewer. The authors also provide an example (from the class of symmetric birth-and-death processes) in answer to a question posed by Fukushima, demonstrating that \(Y\backslash\tilde Y\) need not be of zero capacity.
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 91628
zbMATH Keywords: \(m\)-symmetric Hunt process
zbMATH Keywords: locally compact separable state space
reviewed by: Patrick J. Fitzsimmons
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 91628