Infinite graphs satisfying the Bakry-Emery curvature condition CD(0, n): The modified heat equation and applications to geometric analysis

TL;DR

This paper proves that infinite weighted graphs with non-negative Bakry-Emery curvature satisfy the doubling volume property by establishing a unique solution to a modified heat equation, extending geometric analysis tools to graph settings.

Contribution

It introduces a modified heat equation for graphs satisfying CD(0, n) and proves key estimates, bridging techniques from Riemannian geometry to graph theory.

Findings

Doubling volume property holds for graphs with CD(0, n)

Existence and uniqueness of solutions to the modified heat equation

Li-Yau and Harnack estimates established for the solutions

Abstract

Let G = (V, p, $\mu$) be a (finite or infinite) weighted graph with bounded geometry. Assuming that G satisfies the classical curvaturedimension condition of Bakry-Emery CD(K, n) with K $\ge$ 0 (for the usual Laplacian), we prove that the doubling volume property holds. One of the key points is to establish the existence and uniqueness of solutions of a modified non linear heat equation which replaces the standard one usually used in the case of Riemannian manifolds. Li-Yau and Harnack estimates for the solutions of this modified heat equation are obtained. We also provide explicit examples of Cayley graphs satisfying our assumptions.