Li-Yau type differential Harnack inequalities and applications for nonlocal diffusion equations

• Applicant: Professor Dr. Rico Zacher
• Subject Area: Mathematics
• Term: from 2017 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 355354916

• Final Report Year: 2023

Final Report Abstract

The main subject of the project was the derivation of Li-Yau type differential Harnack inequalities for a wide class of nonlocal diffusion equations including problems on finite and infinite discrete structures (graphs) and space fractional diffusion equations in the Euclidean space. A major difficulty consisted in the fact that for nonlocal operators the classical chain rules fail to hold, which in particular means that the curvature-dimension (CD) inequality based on the Γ-calculus of Bakry and Emery is not suitable in this context. For this reason we have developed an appropriate calculus for nonlocal operators (which are composed of difference terms) and introduced corresponding new CD-conditions. One of the new key ideas here was to replace the quadratic dimension term in the classical Bakry-Emery condition by a more general term involving a so-called CD-function. This new concept turned out to be crucial in case of discrete long-range jump operators, e.g. fractional powers of the discrete Laplacian, where the kernel of the operator determines the CD-function. In the latter example, in contrast to the square function from the classical case, the associated CD-function has a superquadratic behaviour at zero and growths exponentially at infinity. Another important feature in our new CD-inequalities is the prominent role of the function Υ(x) = exp(x) − 1 − x, which substitutes in many places the square function x^2 /2 in the nonlocal setting. By using the new CD-conditions we were able to prove Li-Yau type differential Harnack inequalities for a variety of discrete problems like the heat equation on locally finite graphs and fractional powers of the one-dimensional discrete Laplacian. Staying in the discrete setting, another important result of the project was to identify a CD- inequality, the condition CDΥ (κ, ∞) with curvature parameter κ ∈ R, which serves as a natural analogue of the classical Bakry-Emery condition CD(κ, ∞) in several respects and fits to our calculus developed for the Li-Yau inequalities. In particular, positive curvature bounds in the sense of CDΥ imply the modified logarithmic Sobolev inequality and are preserved under tensorization. They are also compatible with the diffusive setting, in the sense that corresponding hybrid processes enjoy a tensorization property. Surprisingly, it turned out that the fractional Laplacian in Euclidean space does not satisfy the Bakry-Emery condition (and also not the CDΥ -condition) with finite dimension, which considerably complicated the derivation of a corresponding Li-Yau inequality. Therefore we used the completely different approach of reducing the problem to the heat kernel. We established in a very general framework (covering the continuous and discrete setting) a reduction-to-heat-kernel principle, which works for positive solutions to a wide class of nonlocal diffusion equations. The principle says that validity of the Li-Yau inequality for the heat kernel implies the validity of the Li-Yau inequality for any positive solution. Employing certain bounds of the fractional heat kernel and its derivatives we were thus able to prove a Li-Yau inequality for the fractional Laplacian in the continuous setting with a c/t behaviour of the relaxation function. Another important aspect of the project was to use the obtained Li-Yau type inequalities to prove new (parabolic) Harnack inequalities for nonnegative solutions to the described diffusion problems or to find new, much more simple (purely analytic) proofs of the Harnack inequality in cases where it is already known. Concerning time fractional diffusion equations with time order less than one we found the surprising result that the classical parabolic Harnack inequality does not hold in higher dimensions whereas it does hold in the one-dimensional case.

Publications

• Curvature-dimension inequalities for non-local operators in the discrete setting. Calculus of Variations and Partial Differential Equations, 58(5).
  Spener, Adrian; Weber, Frederic & Zacher, Rico
  
• Long-time behavior of non-local in time Fokker–Planck equations via the entropy method. Mathematical Models and Methods in Applied Sciences, 29(02), 209-235.
  Kemppainen, Jukka & Zacher, Rico
  
• On the parabolic Harnack inequality for non-local diffusion equations. Mathematische Zeitschrift, 295(3-4), 1751-1769.
  Dier, Dominik; Kemppainen, Jukka; Siljander, Juhana & Zacher, Rico
  
• The fractional Laplacian has infinite dimension. Communications in Partial Differential Equations, 45(1), 57-75.
  Spener, Adrian; Weber, Frederic & Zacher, Rico
  
• Discrete versions of the Li-Yau gradient estimate. ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 691-744.
  Dier, Dominik; Kassmann, Moritz & Zacher, Rico
  
• The entropy method under curvature-dimension conditions in the spirit of Bakry-Émery in the discrete setting of Markov chains. Journal of Functional Analysis, 281(5), 109061.
  Weber, Frederic & Zacher, Rico
  
• Li–Yau inequalities for general non-local diffusion equations via reduction to the heat kernel. Mathematische Annalen, 385(1-2), 393-419.
  Weber, Frederic & Zacher, Rico
  
• Li-Yau and Harnack inequalities via curvature-dimension conditions for discrete long-range jump operators including the fractional discrete Laplacian. Discrete and Continuous Dynamical Systems, 44(7), 1982-2028.
  Kräss, Sebastian; Weber, Frederic & Zacher, Rico