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

该项目的主要目的是给出一大类非局部扩散方程的Li-Yau型微分Harnack不等式，其中包括无限离散结构(图)上的问题、欧氏空间中涉及分数拉普拉斯算子的方程以及分数动力学问题。一个主要的困难是对于所涉及的非局部算子的经典链规则的失败。该项目的一个关键部分在于证明曲率-维度(CD)不等式的新变体。我们以前的工作表明经典的Gamma演算是不适用的，并建议考虑包含所谓的(非二次)CD函数的CD-不等式，该CD函数依赖于非局部算子REP。底层结构。借助于新的Li-Yau型不等式，我们分别导出了新的Harnack不等式。为这些结果找到新的、更简单的(纯解析的)证明(例如，在空间分数的情况下)。我们还想研究(新的)CD-不等式在关于泛函不等式(Poincare，Sobolev和LOG-Sobolv不等式)的正下曲率下界的应用，这反过来又将导致关于扩散方程解的长期行为的进一步结果。

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.