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**On the closable parts of pre-Dirichlet forms and the fine supports of underlying measures.**
*(English)*
Zbl 0756.60071

Let \(({\mathcal E},{\mathcal F})\) be an either irreducible or transient regular Dirichlet form on \(L^ 2(X;m)\) where \(X\) is a locally compact separable metric space, \(m\in{\mathcal M}\) with \(\text{supp}[m]=X\), and \({\mathcal M}:=\{\nu\mid \nu\) positive Radon measure on \(X\}\). Let \({\mathcal C}\subset{\mathcal F}\) be a dense subalgebra of continuous functions with compact support on \(X\) such that

\(({\mathcal C}.1)\) For every \(K,G\subset X\) with \(K\subset G\), \(K\) compact, \(G\) relatively compact and open, there exists \(w\in{\mathcal C}\) such that \(0\leq w\leq 1\), \(w=1\) on \(K\) and \(w=0\) on \(X\setminus G\).

\(({\mathcal C}.2)\) For every \(\varepsilon>0\) there exists \(\beta_ \varepsilon:\mathbb{R}\to[-\varepsilon,1+\varepsilon]\) such that \(\beta_ \varepsilon(t)=t\) for all \(t\in[0,1]\) and \(0\leq\beta_ \varepsilon(t)- \beta_ \varepsilon(s)\leq t-s\) for \(t>s\) and \(\beta_ \varepsilon(f)\in{\mathcal C}\) whenever \(f\in{\mathcal C}\).

Let \(Cap\) denote the 1-capacity associated with \(({\mathcal E},{\mathcal F})\). Let \({\mathcal M}_ 0:=\{\nu\mid \nu\) charges no Cap-zero set\(\}\), \({\mathcal M}_{00}:=\{\nu\in{\mathcal M}_ 0\mid Cap(X\setminus\tilde S_ \nu)=0\}\), where \(\tilde S_ \nu\) denotes the support of the positive continuous additive functional (for the underlying Hunt process) associated with \(\nu\in{\mathcal M_ 0}\). The first result of this very interesting paper is that each \(\mu\in{\mathcal M}\) has a unique decomposition \[ \mu=\mu_ 0+\mu_ 1,\quad \mu_ 0\in{\mathcal M}_ 0, \quad \mu_ 1=1_ N\cdot\mu\text{ with }Cap(N)=0. \] The main result states then that for \(\mu\in{\mathcal M}\) with \(\text{supp}[\mu]=X\), \[ ({\mathcal E},{\mathcal C})\text{ is closable on }L^ 2(X;\mu)\text{ if and only if }\mu_ 0\in{\mathcal M}_{00}.\tag{1} \] The techniques of the paper combine in a very nice way analytic and probabilistic methods. In particular, the existence of the closable part of \(({\mathcal E},{\mathcal C})\) on \(L^ 2(X;\mu)\) (for arbitrary \(\mu\)) is proved, i.e., the existence of a largest closable form on \(L^ 2(X;\mu)\) dominated by \(({\mathcal E},{\mathcal C})\). Apart from the analytic relevance, (1) is very interesting from a probabilistic point of view since it can be difficult to check whether a measure in \({\mathcal M}_ 0\) belongs to \({\mathcal M}_{00}\). Closability, however, has been studied analytically in an extensive way. The main result of this paper also generalizes a joint result of the reviewer and N. Wielens. It should be noted that using recently developed regularization techniques one can extend the above results to not necessarily regular Dirichlet forms on general state spaces.

\(({\mathcal C}.1)\) For every \(K,G\subset X\) with \(K\subset G\), \(K\) compact, \(G\) relatively compact and open, there exists \(w\in{\mathcal C}\) such that \(0\leq w\leq 1\), \(w=1\) on \(K\) and \(w=0\) on \(X\setminus G\).

\(({\mathcal C}.2)\) For every \(\varepsilon>0\) there exists \(\beta_ \varepsilon:\mathbb{R}\to[-\varepsilon,1+\varepsilon]\) such that \(\beta_ \varepsilon(t)=t\) for all \(t\in[0,1]\) and \(0\leq\beta_ \varepsilon(t)- \beta_ \varepsilon(s)\leq t-s\) for \(t>s\) and \(\beta_ \varepsilon(f)\in{\mathcal C}\) whenever \(f\in{\mathcal C}\).

Let \(Cap\) denote the 1-capacity associated with \(({\mathcal E},{\mathcal F})\). Let \({\mathcal M}_ 0:=\{\nu\mid \nu\) charges no Cap-zero set\(\}\), \({\mathcal M}_{00}:=\{\nu\in{\mathcal M}_ 0\mid Cap(X\setminus\tilde S_ \nu)=0\}\), where \(\tilde S_ \nu\) denotes the support of the positive continuous additive functional (for the underlying Hunt process) associated with \(\nu\in{\mathcal M_ 0}\). The first result of this very interesting paper is that each \(\mu\in{\mathcal M}\) has a unique decomposition \[ \mu=\mu_ 0+\mu_ 1,\quad \mu_ 0\in{\mathcal M}_ 0, \quad \mu_ 1=1_ N\cdot\mu\text{ with }Cap(N)=0. \] The main result states then that for \(\mu\in{\mathcal M}\) with \(\text{supp}[\mu]=X\), \[ ({\mathcal E},{\mathcal C})\text{ is closable on }L^ 2(X;\mu)\text{ if and only if }\mu_ 0\in{\mathcal M}_{00}.\tag{1} \] The techniques of the paper combine in a very nice way analytic and probabilistic methods. In particular, the existence of the closable part of \(({\mathcal E},{\mathcal C})\) on \(L^ 2(X;\mu)\) (for arbitrary \(\mu\)) is proved, i.e., the existence of a largest closable form on \(L^ 2(X;\mu)\) dominated by \(({\mathcal E},{\mathcal C})\). Apart from the analytic relevance, (1) is very interesting from a probabilistic point of view since it can be difficult to check whether a measure in \({\mathcal M}_ 0\) belongs to \({\mathcal M}_{00}\). Closability, however, has been studied analytically in an extensive way. The main result of this paper also generalizes a joint result of the reviewer and N. Wielens. It should be noted that using recently developed regularization techniques one can extend the above results to not necessarily regular Dirichlet forms on general state spaces.

Reviewer: M.Röckner (Bonn)