Functional calculus for Dirichlet forms
Let \(X\) be a not necessarily locally compact, separable metric space and \(m\) a \(\sigma\)-finite Borel measure on \(X\). Given a semi-Dirichlet form \(({\mathcal E},{\mathcal F})\) on \(L^2(X;m)\), let \((T_t)\) and \((G_\alpha)\) be its associated semigroup and resolvent, respectively. The main purpose of this paper is to extend the functional and stochastic calculus already known in the cases of symmetric and non-symmetric Dirichlet forms to the present case. In particular, under a condition of quasi-regularity of \(({\mathcal E},{\mathcal F})\), the notion of the local space is introduced and its characterizations such as identification of the local space of a part space on an \({\mathcal E}\)-quasi-open set are given. Beurling-Deny and LeJan formula and its uniqueness problem related to symmetric quasi-regular Dirichlet forms are discussed. Under the present notion of the locality, the energy measure of the continuous part and its chain rules are given. Ornstein-Uhlenbeck process on an abstract Wiener space and Fleming-Viot process are given as examples.
scientific article; zbMATH DE number 1216678