A characterization of the closable parts of pre-Dirichlet forms by hitting distributions (Q2367026)

Let \(({\mathcal E},{\mathcal F})\) be a \(C_ 0\)-regular Dirichlet form on \(L^ 2 (X,m)\), where \(X\) is a locally compact metric space, and \(m\) is a positive Radon measure. Let \({\mathcal C}\) be a dense subalgebra of the space of continuous functions on \(X\) with compact support satisfying the following two axioms: (C.1) For any compact set \(K\) and relatively compact open set \(G\) with \(K \subset G \subset X\), there exists \(f \in {\mathcal C}\) such that \(0 \leq f \leq 1\), \(f=1\) on \(K\) and \(f=0\) on \(X \backslash G\). (C.2) For any \(\varepsilon>0\) there exists a real function \(\varphi_ \varepsilon\) satisfying that \(\varphi_ \varepsilon(t)=t\) for \(0 \leq t \leq 1\), \(-\varepsilon \leq \varphi_ \varepsilon \leq 1+ \varepsilon\) for any \(t\), and \(0 \leq \varphi_ \varepsilon(t)-\varphi_ \varepsilon(s)\leq t-s\) for \(s \leq t\), and \(\varphi_ \varepsilon(f)\in {\mathcal C}\) whenever \(f \in{\mathcal C}\). Let Cap be the 1-capacity associated with \({\mathcal E}\). The space of positive Radon measures on \(X\) is denoted by \({\mathcal M}\). Let \({\mathcal M}_ 0=\{\nu \in {\mathcal M}:\nu\) charges no \({\mathcal E}_ 1\)-set\} and \({\mathcal M}_{00}=\{\nu \in{\mathcal M}:\text{Cap} (X \backslash \tilde S_ \nu)=0\}\), where \(\tilde S_ \nu\) stands for the quasisupport of \(\nu\) (equivalently, \(\tilde S_ \nu\) is the support of the PCAF \(A^ \nu_ t\) of the underlying \(m\)-symmetric Hunt process \(M=\{\Omega,{\mathcal F}_ t, X_ t,P_ x,x \in X\}\) associated with \(\nu)\). The main result of the paper [\textit{M. Fukushima}, \textit{K. Sato} and \textit{S. Taniguchi}, Osaka J. Math. 28, No. 3, 517-535 (1991; Zbl 0756.60071)] is the following: If \(({\mathcal E},{\mathcal F})\) is either irreducible or transient, then for \(\mu \in {\mathcal M}\) with \(\text{supp} [\mu]=X\), \(({\mathcal E},{\mathcal C})\) is closable on \(L^ 2(X,\mu)\) if and only if \(\mu_ 0 \in {\mathcal M}_{00}\). Here \(\mu_ 0\) is the smooth part of \(\mu\) with respect to Cap. In this paper the question of closability is addressed without the assumption that \(({\mathcal E},{\mathcal F})\) is either irreducible or transient. Firstly, the following decomposition of the state space \(X\) is established: \(X=X^{(c)}+X^{(d)}+N\), where \(N^{(c)}\) (resp. \(X^{(d)})\) is an \({\mathbf M}\)-invariant conservative (resp. dissipative) part of \(X\) and \(N\) is a properly exceptional set. Secondly, for \(\mu \in {\mathcal M}_ 0\), an analytic description of the Dirichlet space of the time changed process \({\mathbf M}^ t=(X_{\tau_ t},P_ x)\), \(x \in \tilde S_ \mu\), is given, where \(\tau_ t\) is the inverse of the PCAF \(A^ \mu_ t\). This result parallels the one in [\textit{K. Kuwae} and \textit{S. Nakao}, ibid. 28, No. 4, 847-865 (1991; Zbl 0764.31005)]. Due to the fact that the form is neither irreducible nor transient, special care has to be taken of the set \(X^{(c)} \backslash B^ \mu\), where \(B^ \mu=\{x \in X:P_ x(A^ \mu_ \infty>0)>0\}\). Thirdly, the closability is established in the following form: for a closed subset \(Y\) of \(X\) let \({\mathcal C}_ Y=\{u \in C_ 0(Y):u=\overline u_ Y\) for some \(\overline u \in {\mathcal C}\}\), and define the pre- Dirichlet form \({\mathcal A}_ Y\) on \({\mathcal C}_ Y\) by \({\mathcal A}_ Y(u,u)=E(H_ Y \overline u, H_ Y \overline u)\), where \(H_ Y \overline u(x)=E_ x[\overline u(X_{\sigma_ Y})]\), \(\sigma_ Y\) being the hitting time to \(Y\). If \(\text{Cap} (Y \backslash \tilde S_{\mu 0})=0\), then \(({\mathcal A}_ Y,{\mathcal C}_ Y)\) is closable on \(L^ 2(Y,\mu)\) and \(Y\cap (X^{(c)} \backslash B^{\mu_ 0}) = \emptyset\) q.e. Conversely, if \(({\mathcal A}_ Y,{\mathcal C}_ Y)\) is closable on \(L^ 2(Y,\mu)\) and \((X^{(c)} \backslash B^{\mu_ 0})=\emptyset\) q.e., then \(\text{Cap} (Y \backslash \tilde S_{\mu_ 0})=0\).