Recurrence and conservativeness of symmetric diffusion processes by Girsanov transformations
type: Item
instance of: scholarly article
title: Recurrence and conservativeness of symmetric diffusion processes by Girsanov transformations
zbMATH Open document ID: 0937.60068
author: Kazuhiro Kuwae
published in: Journal of Mathematics of Kyoto University
publication date: 2000-05-28T00:00:00Z
review text: Let \(X\) be a separable metric space and \(m\) a \(\sigma\)-finite Borel measure on \(X\). Let \(({\mathcal E},{\mathcal F})\) be a quasi-regular Dirichlet space on \(L^2(X;m)\) with associated diffusion process \({\mathbb M}=(X_t,P_x)\) having no killing inside. For a nonnegative quasi-continuous function \(\varphi\) belonging to \({\mathcal F}\) locally in the broad sense, let \(\tau\) be the exit time from \(X_\varphi=\{x \in X:0<\varphi(x)<\infty\}\). Denote by \({\mathbb M}^\varphi=(X_t,P^\varphi_x)\) the process on \(X_\varphi\) transformed by the multiplicative functional \(L^{[\varphi]}_t= \exp[M^{[\log \varphi]}_t-{1 \over 2} \langle M^{[\log \varphi]}\rangle_t] I_{\{t \leq \tau\}}\). The main purpose of this paper is to give the recurrence and conservativeness criteria of \({\mathbb M}^\varphi\) by using the growth of the volume of the ball determined by the intrinsic metric. The criteria given in this paper are generalizations of those given in the \(C_0\)-regular case by \textit{K.-Th. Sturm} [J. Reine Angew. Math. 456, 173-196 (1994; Zbl 0806.53041)] and \textit{H. Ôkura} [in: Dirichlet forms and stochastic processes, 291-303 (1995; Zbl 0840.60071)]. Some examples which cannot be covered by the framework of the regular Dirichlet spaces are also given.
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J57
zbMATH DE Number: 1295315
zbMATH Keywords: recurrence
zbMATH Keywords: conservativeness
zbMATH Keywords: Dirichlet form
reviewed by: Yoichi Oshima
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518212
OpenAlex ID: W1868412793
DOI: 10.1215/KJM/1250518212
description: scientific article; zbMATH DE number 1295315