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Li-Yau and Harnack estimates for nonlocal diffusion problems

Rico Zacher

Abstract

These notes give a brief introduction to differential Harnack inequalities and summarise the main results of the mini-course ``Li-Yau and Harnack estimates for nonlocal diffusion problems'', presented by the author at the Seasonal School on PDEs ``Oscillation Phenomena, PDEs, and Applications: A Comprehensive School in Mathematical Analysis'', held at Ghent University in October 2025.

Li-Yau and Harnack estimates for nonlocal diffusion problems

Abstract

These notes give a brief introduction to differential Harnack inequalities and summarise the main results of the mini-course ``Li-Yau and Harnack estimates for nonlocal diffusion problems'', presented by the author at the Seasonal School on PDEs ``Oscillation Phenomena, PDEs, and Applications: A Comprehensive School in Mathematical Analysis'', held at Ghent University in October 2025.

Paper Structure

This paper contains 5 sections, 5 theorems, 55 equations.

Table of Contents

  1. Introduction: Classical Li--Yau and CD inequality
  2. Nonlocal operators: examples and difficulties
  3. The $CD_\Upsilon$-condition
  4. Long-range jump operators
  5. Reduction to the heat kernel

Key Result

Theorem 4.1

Let the kernel $k$ be as described before. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (6)

  • Definition 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 5.1