Differential Harnack inequalities for non-local diffusion problems
A talk in the Nonlocal Equations: Analysis and Numerics series by
Rico Zacher from Ulm
| Abstract: | I will present recent results on differential Harnack inequalities of Li-Yau
type for certain classes of nonlocal diffusion equations. This includes prob-
lems on infinite discrete structures (graphs) on which arbitrarily long jumps
are possible and problems in Euclidean space with a fractional Laplace op-
erator. 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 ad-
dition 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. This is partly joint work with
S. Kräss, A. Spener and F. Weber. Within the CRC this talk is associated to the project(s): A7 |