On the Structure of the Domain of a Symmetric Jump-type Dirichlet Form

Abstract

We characterize the structure of the domain of a pure jump-type Dirichlet form which is given by a Beurling{Deny formula. In particular, we obtain sufficient conditions in terms of the jumping kernel guaranteeing that the test functions are a core for the Dirichlet form and that the form is a Silverstein extension. As an application we show that for recurrent Dirichlet forms the extended Dirichlet space can be interpreted in a natural way as a homogeneous Dirichlet space. For reflected Dirichlet spaces this leads to a simple purely analytic proof that the active reflected Dirichlet space (in the sense of Chen, Fukushima and Kuwae) coincides with the extended active reflected Dirichlet space.