[Inégalité de Li–Yau et applications sous une condition de courbure-dimension]
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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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
@article{AIF_2017__67_1_397_0,
author = {Bakry, Dominique and Bolley, Fran\c{c}ois and Gentil, Ivan},
title = {The {Li{\textendash}Yau} inequality and applications under a curvature-dimension condition},
journal = {Annales de l'Institut Fourier},
pages = {397--421},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {67},
number = {1},
year = {2017},
doi = {10.5802/aif.3086},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.3086/}
}
TY - JOUR AU - Bakry, Dominique AU - Bolley, François AU - Gentil, Ivan TI - The Li–Yau inequality and applications under a curvature-dimension condition JO - Annales de l'Institut Fourier PY - 2017 SP - 397 EP - 421 VL - 67 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3086/ DO - 10.5802/aif.3086 LA - en ID - AIF_2017__67_1_397_0 ER -
%0 Journal Article %A Bakry, Dominique %A Bolley, François %A Gentil, Ivan %T The Li–Yau inequality and applications under a curvature-dimension condition %J Annales de l'Institut Fourier %D 2017 %P 397-421 %V 67 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3086/ %R 10.5802/aif.3086 %G en %F AIF_2017__67_1_397_0
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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