A note on the volume growth criterion for stochastic completeness of weighted graphs

TL;DR

This paper extends the weak Omori-Yau maximum principle to strongly local Dirichlet forms and uses it to analytically compare stochastic completeness of weighted and metric graphs, providing new proofs for volume growth criteria.

Contribution

It introduces an analytic approach to stochastic completeness comparison, generalizing maximum principles and offering alternative proofs for existing volume growth criteria.

Findings

Analytic comparison of stochastic completeness between weighted and metric graphs

Extension of the weak Omori-Yau maximum principle to strongly local Dirichlet forms

Provision of an alternative analytic proof for volume growth criteria

Abstract

We generalize the weak Omori-Yau maximum principle to the setting of strongly local Dirichlet forms. As an application, we obtain an analytic approach to compare the stochastic completeness of a weighted graph with that of an associated metric graph. This comparison result played an essential role in the volume growth criterion of Folz \cite{FOLZSC}, who first proved it via a probabilistic approach. We also give an alternative analytic proof based on a criterion in Fukushima, Oshima, and Takeda \cite{FOT}.