Upper escape rate of Markov chains on weighted graphs

TL;DR

This paper derives an upper escape rate function for continuous-time symmetric Markov chains on weighted graphs, linking volume growth to escape behavior, and extends related conservativeness results.

Contribution

It introduces a new upper escape rate function based on volume growth and path metrics, extending manifold results to weighted graphs.

Findings

Upper escape rate function expressed via volume growth

Extension of Folz's theorem on conservativeness

Applicable to continuous-time symmetric Markov chains

Abstract

We obtain an upper escape rate function for a continuous time minimal symmetric Markov chain, defined on a locally finite weighted graph. This upper rate function is given in terms of volume growth with respect to an adapted path metric and has the same form as the manifold setting. Our approach also gives a slightly more restrictive form of Folz's theorem on conservativeness as a consequence.