Li-Yau inequalities for general non-local diffusion equations via reduction to the heat kernel

We establish a reduction principle to derive Li-Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li-Yau inequality for positive solutions u to the fractional (in space) heat equation of the form (- Delta)(beta/2)(log u) <= C/t, where beta epsilon (0, 2). We also show that this Li-Yau inequality allows to derive a Harnack inequality. We further illustrate our general result with an example in the discrete setting by proving a sharp Li-Yau inequality for diffusion on a complete graph.