Functional calculus for Dirichlet forms
type: Item
instance of: scholarly article
title: Functional calculus for Dirichlet forms
zbMATH Open document ID: 0917.60068
author: Kazuhiro Kuwae
published in: Osaka Journal of Mathematics
publication date: 1999-07-18T00:00:00Z
review text: Let \(X\) be a not necessarily locally compact, separable metric space and \(m\) a \(\sigma\)-finite Borel measure on \(X\). Given a semi-Dirichlet form \(({\mathcal E},{\mathcal F})\) on \(L^2(X;m)\), let \((T_t)\) and \((G_\alpha)\) be its associated semigroup and resolvent, respectively. The main purpose of this paper is to extend the functional and stochastic calculus already known in the cases of symmetric and non-symmetric Dirichlet forms to the present case. In particular, under a condition of quasi-regularity of \(({\mathcal E},{\mathcal F})\), the notion of the local space is introduced and its characterizations such as identification of the local space of a part space on an \({\mathcal E}\)-quasi-open set are given. Beurling-Deny and LeJan formula and its uniqueness problem related to symmetric quasi-regular Dirichlet forms are discussed. Under the present notion of the locality, the energy measure of the continuous part and its chain rules are given. Ornstein-Uhlenbeck process on an abstract Wiener space and Fleming-Viot process are given as examples.
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 31C25
zbMATH DE Number: 1216678
zbMATH Keywords: Dirichlet form
zbMATH Keywords: stochastic calculus
zbMATH Keywords: quasi-regular
reviewed by: Yoichi Oshima
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1216678