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

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}.