Nowhere Radon smooth measures, perturbations of Dirichlet forms and singular quadratic forms

Abstract

We expose some new results concerning Dirichlet forms on locally compact spaces, associated Markov processes, and potential theory. In particular to any regular Dirichlet form there exist nowhere Radon smooth measures provided each single-point set is a set of zero capacity. We give examples of such measures of the type of generalized Schrodinger operators. We also present an approximation result of smooth measures by smooth measures in a Kato class considered before in connection with perturbations of Schrodinger operators. We also study perturbations of regular Dirichlet forms by symmetric bilinear forms given by differences of smooth measures, providing in particular new criteria for closability and form cores.