Coverings and the heat equation on graphs: stochastic incompleteness, the Feller property and uniform transience

TL;DR

This paper investigates how properties related to the heat equation, such as stochastic incompleteness and the Feller property, are preserved under regular coverings of graphs and manifolds, establishing key equivalences and new conditions.

Contribution

It proves the equivalence of stochastic incompleteness between a base space and its cover for graphs and manifolds, and introduces new criteria for the Feller property on graphs.

Findings

Stochastic incompleteness is equivalent on base and covering spaces.

New conditions for the Feller property on graphs are established.

Connections between heat equation properties and coverings are clarified.

Abstract

We study regular coverings of graphs and manifolds with a focus on properties of the heat equation. In particular, we look at stochastic incompleteness, the Feller property and uniform transience; and investigate the connection between the validity of these properties on the base space and its covering. For both graphs and manifolds, we prove the equivalence of stochastic incompleteness of the base and that of its cover. Along the way we also give some new conditions for the Feller property to hold on graphs.