Time changes of symmetric Markov processes and a Feynman-Kac formula
Let X be an m-symmetric right Markov process associated with the (not necessarily regular) Dirichlet spce (\({\mathcal D}(X),{\mathcal A})\). In this context, the author proves the Feynman-Kac formula; if \(\Gamma\) is a continuous additive functional (CAF) with Revuz measure \(\nu\), then the Dirichlet space (\({\mathcal D}(X^{\Gamma}),{\mathcal A}^{\Gamma})\) of the subprocess \(X^{\Gamma}\) of X by the multiplicative functional exp(- \(\Gamma)\) is given by \[ {\mathcal D}(X^{\Gamma})={\mathcal D}(X)\cap L^ 2(\nu)\quad and\quad {\mathcal A}^{\Gamma}(u,u)={\mathcal A}(u,u)+\| u\|^ 2. \] A similar result has been obtained by Nakao [Feynman-Kac formula in Dirichlet space; to appear] for regular Dirichlet spaces. The characterization of the extended Dirichlet space (\({\mathcal D}_ e(\tilde X),\tilde {\mathcal A})\) of the time changed process \(\tilde X\) of X by a CAF \({\mathcal B}\) with Revuz measure \(\mu\) was given by \textit{M. L. Silverstein} [Symmetric Markov processes. Lecture Notes Math. 426 (1974; Zbl 0296.60038)], \textit{M. Fukushima} [Dirichlet forms and Markov processes. (1980; Zbl 0422.31007)] and the reviewer [Potential theory and related fields, Proc. Symp., Kyoto/Jap. 1986, RIMS Kokyuroku 610, 131-136 (1987; Zbl 0663.60061)]. If (\({\mathcal D}(X),{\mathcal A})\) is a transient regular Dirichlet space, then (\({\mathcal D}_ e(\tilde X),\tilde {\mathcal A})\) is characterized as the orthogonal complement of \(\{\) \(u\in {\mathcal D}_ e(X):\) quasi-cont. mod. of \(u=0\) q.e. on supp \(B\}\) in (\({\mathcal D}_ e(X),{\mathcal A}).\) In this paper, by using the Feynman-Kac formula, the author reduces the recurrent processes to transient ones and gives a complete characterization of the time changed Dirichlet spaces.
scientific article; zbMATH DE number 4119363
scientific article