In the celebrated paper [2], Li and Yau proved the parabolic Harnack inequality for Riemannian manifolds with Ricci curvature bounded from below. The key step in their proof was a completely new type of Harnack estimate, namely a pointwise gradient estimate, called {\em differential Harnack inequality}, which, by integration along a path, yields the classical parabolic Harnack estimate. If one tries to apply this method to discrete structures (graphs) one is faced with two big obstacles.

The main difficulty is that the chain rule for the Laplace operator fails on graphs. Another problem is that in the graph setting, it is a priori not so clear how to define a proper notion of curvature, or more precisely the concept of lower bounds for the Ricci curvature.

A first successful attempt to circumvent these difficulties was made in the recent paper [1] and is based on the square-root approach. In my talk, I will present a different approach, which, as in the classical case ([2]), leads to logarithmic Li-Yau inequalities, and also significantly improves the results from [1]. This is joint work with D.\ Dier

(Ulm) and M.\ Kassmann (Bielefeld). Most of the presented results are from [3]. I will also discuss some new results on discrete diffusion equations with long range jumps.

References

[1] F. Bauer, P. Horn, Y. Lin, G. Lippner, D. Mangoubi, S.-T. Yau: Li-Yau inequality on graphs. J. Differential Geom. 99 (2015), 359-405.

[2] P. Li, S.-T. Yau: On the parabolic kernel of the Schrödinger operator. Acta. Math. 156 (1986), 153-201.

[3] D. Dier, M. Kassmann, R. Zacher: Discrete versions of the Li-Yau gradient estimate, Preprint 2017, available at arxiv.

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