Li-Yau inequalities for general non-local diffusion equations via reduction to the heat kernel

TL;DR

This paper introduces a general reduction method to derive Li-Yau inequalities for non-local diffusion equations, covering both discrete and continuous cases, and applies it to solve a longstanding open problem.

Contribution

The authors develop a novel reduction principle based on heat kernel representations to establish Li-Yau inequalities for non-local diffusion equations, including fractional heat equations.

Findings

Derived Li-Yau inequality for fractional heat equations with $(- riangle)^{eta/2}( ext{log} u)\, extless C/t$

Solved a long-standing open problem in fractional diffusion by establishing a Li-Yau inequality

Proved a sharp Li-Yau inequality for diffusion on a complete graph

Abstract

We establish a reduction principle to derive Li-Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li-Yau inequality for positive solutions $u$ to the fractional (in space) heat equation of the form $(-\Delta)^{\beta/2}(\log u)\leq C/t$, where $\beta\in (0,2)$. We also illustrate our general result with an example in the discrete setting by proving a sharp Li-Yau inequality for diffusion on a complete graph.