The Li-Yau inequality and applications under a curvature-dimension condition

Dominique Bakry (1) , François Bolley (2) , Ivan Gentil (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 ICJ - Institut Camille Jordan

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality implies all classical Li-Yau type inequalities known to us. Moreover, on a Riemannian manifold, it proves to be equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, and giving new subsequents bounds on the heat kernel of the semigroup.

Mots clés: Li-Yau inequality; Harnack inequality; heat kernel bounds; curvature-dimension condition; logarithmic Sobolev inequality

Domaines: Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Dominique Bakry, François Bolley, Ivan Gentil. The Li-Yau inequality and applications under a curvature-dimension condition. 2014. ⟨hal-01094046v1⟩