Global properties of Dirichlet forms on discrete spaces

TL;DR

This paper explores the global properties of Dirichlet forms on discrete spaces, including recurrence and stochastic completeness, using functional analytic methods and relating these to spectral theory and Markov chains.

Contribution

It provides new characterizations of recurrence for Dirichlet forms on discrete spaces solely through functional analytic techniques.

Findings

Characterization of recurrence via Dirichlet form methods

Comparison of Dirichlet form recurrence with Markov chain recurrence

Analysis of the relation between global properties and spectral theory

Abstract

We provide an introduction to Dirichlet forms on discrete spaces and study their global properties such as recurrence, stochastic completeness and regularity of the Neumann form. In this setting we compare the notion of a recurrent Dirichlet form and a recurrent discrete time Markov chain of a given graph. We prove several known and several new characterizations of recurrence by using functional analytic Dirichlet form methods only. Finally, we compare all the mentioned global properties and discuss their relation to spectral theory.