Prof. Dr. Rico Zacher, Universität Ulm

I will present recent results on differential Harnack inequalities of Li-Yau type for certain classes of nonlocal diffusion equations. This includes problems on infinite discrete structures (graphs) on which arbitrarily long jumps are possible and problems in Euclidean space with a fractional Laplace operator. One of the main difficulties is that the classical chain rule is not valid for the nonlocal operators under consideration. Additionally, if one wants to adopt Li and Yau’s approach from their famous 1986 paper (Acta. Math.), new curvature-dimension (CD) inequalities are required, since the classical Bakry-Emery condition based on the Gamma calculus is no longer suitable. This also touches on the fundamental question of how to define lower curvature bounds on discrete structures in a meaningful way. In addition to the approach using CD inequalities, I will present another method which is based on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel.