Dirichlet forms and stochastic completeness of graphs and subgraphs

TL;DR

This paper investigates the properties of Laplacians on graphs using Dirichlet forms, providing conditions for selfadjointness, characterizing stochastic completeness, and exploring the relationship between subgraph and graph stochastic properties.

Contribution

It introduces a geometric criterion for essential selfadjointness, explicitly determines semigroup generators on all  spaces, and generalizes stochastic completeness results for graph Laplacians.

Findings

Established a sufficient geometric condition for essential selfadjointness.

Explicitly characterized generators of semigroups on all  spaces.

Generalized stochastic completeness criteria for graphs and subgraphs.

Abstract

We study Laplacians on graphs and networks via regular Dirichlet forms. We give a sufficient geometric condition for essential selfadjointness and explicitly determine the generators of the associated semigroups on all $\ell^p$, $1\leq p < \infty$, in this case. We characterize stochastic completeness thereby generalizing all earlier corresponding results for graph Laplacians. Finally, we study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph.