JournalsprimsVol. 48, No. 1pp. 1–20

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

  • René L. Schilling

    Technische Universität Dresden, Germany
  • Toshihiro Uemura

    Kansai University, Osaka, Japan
On the Structure of the Domain of a Symmetric Jump-type Dirichlet Form cover
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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 sucient 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.

Cite this article

René L. Schilling, Toshihiro Uemura, On the Structure of the Domain of a Symmetric Jump-type Dirichlet Form. Publ. Res. Inst. Math. Sci. 48 (2012), no. 1, pp. 1–20

DOI 10.2977/PRIMS/58