Measures and Dirichlet forms under the Gelfand transform

TL;DR

This paper explores how Dirichlet forms on measurable spaces can be transformed into regular forms on locally compact spaces using advanced mathematical tools, enabling the construction of associated Markov processes.

Contribution

It demonstrates a method to convert Dirichlet forms into regular forms on locally compact spaces, establishing the existence of related Hunt processes and Markov processes for resistance forms.

Findings

Any Dirichlet form can be transformed into a regular form on a locally compact space.

Existence of Hunt processes on the Stone-ech compactification.

Associated Markov processes for separable resistance forms.

Abstract

Using the standard tools of Daniell-Stone integrals, Stone-\v{C}ech compactification and Gelfand transform, we discuss how any Dirichlet form defined on a measurable space can be transformed into a regular Dirichlet form on a locally compact space. This implies existence, on the Stone-\v{C}ech compactification, of the associated Hunt process. As an application, we show that for any separable resistance form in the sense of Kigami there exists an associated Markov process.