Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms
The theory of regular Dirichlet forms \((E,{\mathcal F})\) associated with a locally compact separable metric space \({\mathcal X}\) and a positive Radon measure \(m\) such that \(\text{supp} [m]={\mathcal X}\) is a well developed subject, both from the potential analytic and the probabilistic point of view. The theory has turned out, in the last 15 years, to be particularly suited for applications in quantum theory. In this field, but also in other contexts, there is the necessity of studying certain generalized functionals of the processes (of Feynman-Kac type), corresponding to singular perturbations of a given Dirichlet form. Many of the discussions have been concerned with functionals associated with measures in the so- called Kato class. They are particular cases of smooth measures for the Dirichlet form associated with the Laplacian. It is natural to ask one- selves what happens if one tries to carry through similar constructions using an arbitrary smooth measure associated with a general (regular) Dirichlet form. In the present paper we initiate such study. We give results on the structure of the space \(S\) of all smooth measures associated with a given Dirichlet form, as well as on their associated Feynman-Kac functionals and generalized resolvents. By so doing we prepare our way for a systematic study of singular perturbations of Dirichlet forms, to be carried through in a successive publication [authors, J. Funct. Anal. 99, No. 2, 332-356 (1991; Zbl 0743.60071)].
singular perturbations of Dirichlet forms
scientific article; zbMATH DE number 219872