Li-Yau inequality under $CD(0,n)$ on graphs

Abstract

We introduce a modified non-linear heat equation ${\partial }_{t}u=\Delta u+\Gamma u$ as a substitute of $\log P_{t}f$ where $P_t$ is the heat semigroup. We prove an exponential decay of $\Gamma u$ under the Bakry Emery curvature condition $CD(K,\infty )$ and prove the Li-Yau inequality $-\Delta u_t\le \frac{n}{2t}$ under the Bakry Emery curvature condition $CD(0,n)$. From this, we deduce the volume doubling property which solves a major open problem in discrete Ricci curvature. As an application, we show that there exist no expander graphs satisfying $CD(0,n)$.

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