Probability Seminar: Matthew Folz

We analyze stochastic completeness, or non-explosiveness, of the variable-speed
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.