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

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.

Bakry-Émery curvature on graphs as an eigenvalue problem

“…In recent years, the discrete Bakry-Émery theory on graphs has become an active emerging research field. There are a growing number of articles investigating this theory, see e.g., [7,8,9,10,11,12,14,15,17,20,21,22,23,24,26,27,28,29,31,32,33,34,35,37,38,39,41,42,44,45,47,48,49,52]. Let us mention here important related works on non-linear discrete curvature dimension inequalities, see e.g., [4,13,18,19,43].…”

Inner-Outer Curvatures, Ricci-Ollivier Curvature and Volume Growth of Graphs

“…We end this subsection showing that, in general, even if two graphs have the same Ollivier curvature k(r) for every r it is not possible to conclude that they have the same volume growth. This is somehow interesting, being a sign that, even for birth-death chains, the Ollivier curvature is not capable of controlling the volume of spheres (see Proposition 3.5 and Remark 3.6 in [6] for similar results).…”