VOLUME GROWTH AND STOCHASTIC COMPLETENESS OF GRAPHS
MATTHEW FOLZ
We analyze stochastic completeness, or non-explosiveness, of the variablespeed random walk (VSRW) on weighted graphs. We prove a criterion relating volume growth in an adapted metric to stochastic completeness
of the VSRW. This criterion is analogous to the optimal result for Riemannian manifolds and is shown to be sharp. The proof is accomplished
through the construction of a Brownian motion on a metric graph which
behaves similarly to the VSRW under consideration. Results of Sturm on
stochastic completeness for local Dirichlet spaces are then applicable to
this Brownian motion, and non-explosiveness of the Brownian motion is
shown to imply non-explosiveness of the VSRW.