Time changes in Dirichlet space theory
Let \(X\) be an irreducible \(C_ 0\)-regular \(m\)-symmetric Hunt process on a locally compact separable state space, and \(({\mathcal E},{\mathcal F})\) denote the associated Dirichlet space on \(L^ 2(m)\). Let \(A=(A_ t)_{t\geq0}\) be a positive continuous additive functional of \(X\) with Revuz measure \(\mu\) and fine support \(\tilde Y\). It is well-known that the time-changed process \(\tilde X_ t:=X_{A^{-1}(t)}\) is a \(\mu\)-symmetric strong Markov process with natural state space \(\tilde Y\). Let \(Y\) denote the topological support of \(\mu\). Then \(\tilde Y\backslash Y\) is of zero capacity and \(\mu(Y\backslash\tilde Y)=0\), so \(\tilde X\) may be viewed as a \(\mu\)-symmetric process on \(Y\). The authors show that after modification on an exceptional set, \(\tilde X\) becomes a Hunt process. Moreover, the associated Dirichlet space is \(C_ 0\)-regular if \(Y\) is taken as the base space, and the authors identify this Dirichlet space explicitly. These results extend and complete work of Fukushima, Silverstein, Oshima, and the reviewer. The authors also provide an example (from the class of symmetric birth-and-death processes) in answer to a question posed by Fukushima, demonstrating that \(Y\backslash\tilde Y\) need not be of zero capacity.
scientific article; zbMATH DE number 91628