Stochastic completeness of graphs: bounded Laplacians, intrinsic metrics, volume growth and curvature

TL;DR

This paper surveys the concept of stochastic completeness in graphs, exploring how intrinsic metrics, volume growth, and curvature influence this property, and clarifies differences between continuous and discrete cases.

Contribution

It provides a comprehensive overview of stochastic completeness, highlighting the role of intrinsic metrics and curvature criteria, and resolves discrepancies between continuous and discrete frameworks.

Findings

Intrinsic metrics help unify stochastic completeness criteria.

Curvature conditions are crucial for establishing stochastic completeness.

Weakly spherically symmetric graphs demonstrate the sharpness of the results.

Abstract

The goal of this article is to survey various results concerning stochastic completeness of graphs. In particular, we present a variety of formulations of stochastic completeness and discuss how a discrepancy between uniqueness class and volume growth criteria in the continuous and discrete settings was ultimately resolved via the use of intrinsic metrics. Along the way, we discuss some equivalent notions of boundedness in the sense of geometry and of analysis. We also discuss various curvature criteria for stochastic completeness and discuss how weakly spherically symmetric graphs establish the sharpness of results.