Stochastic Completeness of Graphs

TL;DR

This thesis investigates the conditions under which the heat kernel on infinite graphs remains stochastically complete, providing criteria based on vertex valence and spectral properties of the graph Laplacian.

Contribution

It introduces a sufficient condition for stochastic completeness related to vertex valence and demonstrates the optimality of this condition through specific tree examples.

Findings

A sufficient condition for stochastic completeness based on maximum valence.

Optimality of the valence condition shown via tree analysis.

Lower bounds on the spectrum of the graph Laplacian established.

Abstract

In this thesis, we analyze the stochastic completeness of a heat kernel on graphs which is a function of three variables: a pair of vertices and a continuous time, for infinite, locally finite, connected graphs. For general graphs, a sufficient condition for stochastic completeness is given in terms of the maximum valence on spheres about a fixed vertex. That this result is optimal is shown by studying a particular family of trees. We also prove a lower bound on the bottom of the spectrum for the discrete Laplacian and use this lower bound to show that in certain cases the Laplacian has empty essential spectrum.