A study on Markovian maximality, change of probability and regularity
type: Item
instance of: scholarly article
title: A study on Markovian maximality, change of probability and regularity
zbMATH Open document ID: 0813.60070
DOI: 10.1007/BF01049810
author: Shiqi Song
published in: Potential Analysis
publication date: 1995-05-30T00:00:00Z
review text: Let \(E\) be a separable Banach space for which there exists a separable Hilbert space \(H\) such that \(E' \subset H \subset E\) densely and continuously and let \(m\) be a bounded Borel measure on \(E\). Choose an orthonormal basis \(K\) of \(H\). Denote by Ext the family of conservative Dirichlet forms \({\mathcal E}\) on \(L^ 2(E;m)\) such that \(FC_ b^ \infty (K)\) is contained in the domain of the (extended) generator of \({\mathcal E}\) and the caré du champ operator \(\Gamma\) is given by \(\Gamma (u,v) = \sum_{k \in K} {\partial u \over \partial k} {\partial v \over \partial k}\) for \(u,v \in FC^ \infty_ b (K)\). The first assertion of this paper is the existence of the maximal element of Ext relative to the ordering \({\mathcal E} \leq {\mathcal E}'\) if \(D({\mathcal E}) \subset {\mathcal D} ({\mathcal E}')\) and \({\mathcal E} (u,u) \geq {\mathcal E}'(u,u)\) for \(u \in D ({\mathcal E})\). This result has also been given by \textit{S. Albeverio}, \textit{S. Kusuoka} and \textit{M. Röckner} [J. Lond. Math. Soc., II. Ser. 42, No. 1, 122-136 (1990; Zbl 0667.31010)]. Secondly, assume that there exists a symmetric diffusion process \((X_ t, P_ x)\) associated with the maximal element \({\mathcal E}_ K\) of Ext. Take a positive function \(\varphi \in D({\mathcal E})\) and put \(\mu = \varphi^ 2 \cdot m\). The corresponding family Ext defined by \(\mu\) is denoted by \(\text{Ext}_ \mu\). Let \({\mathcal E}_ \varphi \) be the Dirichlet form associated with the \(\mu\)-symmetric diffusion process \((Y_ t, Q_ x)\) obtained from \(X_ t\) by the martingale transformation corresponding to \(L_ t = \exp (M_ t^{\log \varphi} - {1 \over 2} \langle M^{\log \varphi} \rangle_ t)\). Using, effectively, the relation \(U_ \lambda f = R_ \lambda f + 2U_ \lambda [\Gamma (R_ \lambda f, \log \varphi)]\) between the resolvents \(R_ \lambda\) and \(U_ \lambda\) of \(X_ t\) and \(Y_ t\), under the assumption of well-admissibility of each element of \(K\), the author proves that \({…
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 696922
zbMATH Keywords: Markovian extension
zbMATH Keywords: maximality
zbMATH Keywords: martingale transformation
zbMATH Keywords: conservative Dirichlet forms
zbMATH Keywords: symmetric diffusion process
zbMATH Keywords: Gaussian measure
zbMATH Keywords: Caméron-Martin space
reviewed by: Yoichi Oshima
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 696922