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

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The main objective of the project is to derive Li-Yau type differential Harnack inequalities for a wide class of non-local diffusion equations which includes problems on infinite discrete structures (graphs) where arbitrary long jumps are possible, equations in Euclidean space involving a fractional Laplacian and problems with fractional dynamics. A major difficulty is the failure of the classical chain rule for the involved non-local operators. A crucial part of the project consists in proving new variants of the curvature-dimension (CD) inequality. Our previous work shows that the classical Gamma-calculus is not suitable and suggests to consider CD-inequalities involving a so-called (non-quadratic) CD-function, which depends on the non-local operator resp. the underlying structure. By means of the new Li-Yau type inequalities we want to derive new Harnack inequalities resp. find new and much simpler (purely analytic) proofs for such results (e.g. in the space-fractional case). We also want to study applications of the (new) CD-inequalities in case of a positive lower curvature bound with regard to functional inequalities (Poincare, Sobolev and log-Sobolev inequalites), which in turn will lead to further results on the long-time behaviour of the solutions to the diffusion equations.