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arXiv · 2507.19235

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

Emmanuel Russ·Hervé Pajot
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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.

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Emmanuel Russ, Hervé Pajot. 2025-07-25. Infinite graphs satisfying the Bakry-Emery curvature condition CD(0, n): The modified heat equation and applications to geometric analysis. https://arxiv.org/abs/2507.19235

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