The Li–Yau inequality and applications under a curvature-dimension condition
[Inégalité de Li–Yau et applications sous une condition de courbure-dimension]
Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 397-421.

Nous obtenons une inégalité de type Li–Yau pour un semi-groupe de Markov général, sous une condition de courbure-dimension. A notre connaissance, cette nouvelle inégalité renforce toutes les inégalités de ce type. Sur une variété riemannienne, elle est équivalente à une nouvelle inégalité de Harnack parabolique, en courbure positive ou négative, et induit des bornes pertinentes sur le noyau de la chaleur associé. En courbure positive, elle permet d’atteindre des bornes ultracontractives par une méthode directe et robuste.

We prove a global Li–Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li–Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequent bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.

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DOI : 10.5802/aif.3086
Classification : 58J35, 46-XX, 60H15
Keywords: Li–Yau inequality, Harnack inequality, heat kernel bounds, Ricci curvature.
Mot clés : inégalité de Li–Yau, inégalité de Harnack, noyaux de la chaleurs, courbure de Ricci
Bakry, Dominique 1 ; Bolley, François 2 ; Gentil, Ivan 3

1 Institut de mathématiques de Toulouse UMR CNRS 5219 118, route de Narbonne 31062 Toulouse Cedex 9 (France)
2 Laboratoire de probabilités et modèles aléatoires UMR CNRS 7599 4, place Jussieu 75252 Paris Cedex 05 (France)
3 Institut Camille Jordan UMR CNRS 5208 Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918 69622 Villeurbanne (France)
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Bakry, Dominique; Bolley, François; Gentil, Ivan. The Li–Yau inequality and applications under a curvature-dimension condition. Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 397-421. doi : 10.5802/aif.3086. http://www.numdam.org/articles/10.5802/aif.3086/

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