Quasi-regular Dirichlet forms and Markov processes
type: Item
instance of: scholarly article
title: Quasi-regular Dirichlet forms and Markov processes
zbMATH Open document ID: 0777.60072
published in: Journal of Functional Analysis
publication date: 1993-06-29T00:00:00Z
review text: A fundamental result of \textit{M. Fukushima} [Dirichlet forms and Markov processes (1880; Zbl 0422.31007)] says that there exists uniquely (up to q.e. equivalence) a Hunt process associated with a regular Dirichlet form \(({\mathcal E},{\mathcal F})\) on a locally compact space \(X\). But the converse assertion is not true in general. The main result of this paper is to give an analytic characterization of the (not necessarily symmetric) Dirichlet form on a metrizable Lusin state space \(X\) which is associated with a dual pair of special standard (or right) processes \((X,\widehat X)\). More precisely, \(({\mathcal E},{\mathcal F})\) is called quasi-regular if there exists an \({\mathcal E}\)-nest of compact subsets and an \(\widetilde {\mathcal E}_ 1^{1/2}\)-dense subset of \(D({\mathcal E})\) whose elements have \({\mathcal E}\)-q.c. versions, where \(\widetilde{\mathcal E} (u,v)={1\over 2}({\mathcal E}(u,v)+{\mathcal E}(v,u))\). Then a necessary and sufficient condition on \(({\mathcal E},{\mathcal F})\) for the association of the special standard pair \((X,\widehat X)\) is the quasi-regularity. There is a work of \textit{Y. Le Jan} [J. Math. Soc. Jap. 35, 37-42 (1983; Zbl 0528.60070)] in this connection.
reviewed by: Yoichi Oshima
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 31C25
zbMATH DE Number: 220689
zbMATH Keywords: right process
zbMATH Keywords: Hunt process
zbMATH Keywords: Dirichlet form
zbMATH Keywords: quasi-regularity
author: S. Albeverio
author: Zhi-Ming Ma
author: Michael Röckner
MaRDI profile type: Publication
full work available at URL: jfan.1993.1007
OpenAlex ID: W2025676588
DOI: 10.1006/JFAN.1993.1007
description: scientific article; zbMATH DE number 220689