A study on Markovian maximality, change of probability and regularity
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 \({\mathcal E}_ \varphi \in \text{Ext}_ \mu\) and \({\mathcal E}_ \varphi (u,v) = \int \sum_{k \in K} {\partial u \over \partial k} {\partial v \over \partial k} d \mu\). Finally, the regularity of \({\mathcal E}_ K\) or the uniqueness of the element in Ext for the Gaussian measure \(m\) and the Caméron-Martin space \(H\) is shown. This generalizes the result of \textit{M. Takeda} [J. Math. Soc. Japan 44, No. 1, 113-130 (1992)].
scientific article; zbMATH DE number 696922
scientific article