Global properties of Dirichlet forms in terms of Green's formula

TL;DR

This paper investigates global properties of Dirichlet forms, such as uniqueness, stochastic completeness, and recurrence, using Green's formula and boundary term analysis across various mathematical structures.

Contribution

It provides a unified characterization of these properties via boundary term vanishing, extending results to graphs, manifolds, and metric graphs.

Findings

Characterization of Dirichlet form properties through Green's formula

Extension of results to general and regular Dirichlet forms

Explicit determination of operators in graphs and manifolds

Abstract

We study global properties of Dirichlet forms such as uniqueness of the Dirichlet extension, stochastic completeness and recurrence. We characterize these properties by means of vanishing of a boundary term in Green's formula for functions from suitable function spaces and suitable operators arising from extensions of the underlying form. We first present results in the framework of general Dirichlet forms on $\sigma$-finite measure spaces. For regular Dirichlet forms our results can be strengthened as all operators from the previous considerations turn out to be restrictions of a single operator. Finally, the results are applied to graphs, weighted manifolds, and metric graphs, where the operators under investigation can be determined rather explicitly.