A Note on the Volume Growth Criterion for Stochastic Completeness of Weighted Graphs

摘要

This paper gives complementary results of Folz (Trans Am Math Soc, 2013). We first 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 (Trans Am Math Soc, 2013), who first proved it via a probabilistic approach. We also give an alternative analytic proof based on a criterion in Fukushima et al. (1994).