Closability, regularity, and approximation by graphs for separable bilinear forms

TL;DR

This paper establishes a connection between certain quadratic forms and regular Dirichlet forms on metric spaces, and demonstrates approximation of Dirichlet forms by finite graph-based energy forms.

Contribution

It proves a natural isomorphism between subspaces of quadratic form domains and cores of regular Dirichlet forms, and shows approximation of Dirichlet forms by finite weighted graphs.

Findings

Subspace of quadratic form domain is isomorphic to a Dirichlet form core.

Dirichlet forms can be approximated by finite graph energy forms.

Approximation is in the sense of Mosco convergence.

Abstract

We consider a countably generated and uniformly closed algebra of bounded functions. We assume that there is a lower semicontinuous, with respect to the supremum norm, quadratic form and that normal contractions operate in a certain sense. Then we prove that a subspace of the effective domain of the quadratic form is naturally isomorphic to a core of a regular Dirichlet form on a locally compact separable metric space. We also show that any Dirichlet form on a countably generated measure space can be approximated by essentially discrete Dirichlet forms, i.e. energy forms on finite weighted graphs, in the sense of Mosco convergence, i.e. strong resolvent convergence.