On time change of symmetric Markov processes
Let \(M:=(\Omega,{\mathcal B},X_ t,P_ z)\) be an m-symmetric Markov process associated with an irreducible regular Dirichlet space (\({\mathcal E},{\mathcal F})\) on \(L^ 2(X;m)\), where m is a reference measure on the locally compact metric space X. Let A be the positive continuous additive functional associated with a positive Radon measure \(\mu\) on X (not charging sets of capacity zero) and put \[ (1)\quad Y=X(A^{-1}(t)). \] The purpose of this paper is to characterize the extended Dirichlet space (\({\mathcal E}^{\mu},{\mathcal F}^{\mu}_ e)\) associated with the time changed process \(M^{\mu}=(\Omega,{\mathcal B},Y_ t,P_ z).\) Main results are the following: Suppose that M is recurrent in the sense of Harris and let \(\gamma\) be the restriction operator to the support Y of A, then \[ {\mathcal F}^{\mu}_ e={\mathcal H}^ Y\text{ and } {\mathcal E}^{\mu}(\gamma u,\gamma u)={\mathcal E}(u,u),\quad \forall u\in {\mathcal H}^ Y, \] where \({\mathcal H}^ Y\) is defined as the orthogonal complement of \({\mathcal F}_{X-Y}:=\{u\in {\mathcal F}_ s|\) \(u=0\) a.e. in \(Y\}\) in \({\mathcal F}_ s\) (Section 3). In the last Section 4 the author shows under the assumptions \(``X=Y''\) and N has additional finiteness conditions that \((\xi^{\mu},{\mathcal F}^{\mu})\) is regular for general Dirichlet spaces. The additional conditions mentioned above are stated as: either \(| \mu -m|\) is finite or has finite energy integral.
scientific article; zbMATH DE number 4190864