Li–Yau inequality on finite graphs via non-linear curvature dimension conditions

By Florentin Münch. Journal des Mathématiques Pures et Appliquées, 2014

66 citations recorded by the source corpus.

We introduce a new version of a curvature-dimension inequality for non-negative curvature. We use this inequality to prove a logarithmic Li-Yau inequality on finite graphs. To formulate this inequality, we introduce a non-linear variant of the calculus of Bakry and \'Emery. In the case of manifolds, the new calculus and the new curvature-dimension inequality coincide with the common ones. In the case of graphs, they coincide in a limit. In this sense, the new curvature-dimension inequality gives a more general concept of curvature on graphs and on manifolds. We show that Ricci-flat graphs have a non-negative curvature in this sense. Moreover, a variety of non-logarithmic Li-Yau type gradient estimates can be obtained by using the new Bakry-\'Emery type calculus. Furthermore, we use these Li-Yau inequalities to derive Harnack inequalities on graphs.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Genetics

View Paper: https://www.semanticscholar.org/paper/6aeb662e89606f01e85d3323995df7c8d0e18f1f

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In Applied Mathematics

1. Norbert Peyerimhoff
2. Shiping Liu
3. Florentin Münch
4. Ben Andrews

In Astronomy and Astrophysics

1. Norbert Peyerimhoff
2. Shiping Liu
3. Florentin Münch

In Genetics

1. Norbert Peyerimhoff
2. Shiping Liu
3. Florentin Münch

Citations in the corpus

• Bakry–Émery curvature and diameter bounds on graphs (2016, background): “For that reason, there were introduced various non-linear curvature dimension conditions (namely CDE, CDE′, CDψ) closely related to the classical linear CD condition [3, 17].”