Publishing house / Journals and Serials / Dissertationes Mathematicae / All issues

Abstract

We provide an introduction to Dirichlet forms on discrete spaces and study their global properties including recurrence, stochastic completeness and the 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 above mentioned global properties and discuss their relation to spectral theory.