Stochastically Incomplete Manifolds and Graphs

TL;DR

This paper surveys geometric conditions leading to stochastic incompleteness in diffusion processes on manifolds and graphs, providing complete characterizations for symmetric graphs and highlighting differences from Riemannian manifolds.

Contribution

It offers a complete characterization of stochastic incompleteness for spherically symmetric graphs and presents new examples of such graphs with polynomial volume growth.

Findings

Complete characterization for spherically symmetric graphs

Existence of stochastically incomplete graphs with polynomial volume growth

Differences between manifolds and graphs in stochastic incompleteness

Abstract

We survey geometric properties which imply the stochastic incompleteness of the minimal diffusion process associated to the Laplacian on manifolds and graphs. In particular, we completely characterize stochastic incompleteness for spherically symmetric graphs and show that, in contrast to the case of Riemannian manifolds, there exist examples of stochastically incomplete graphs of polynomial volume growth.