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Cordero-Erausquin, Dario; McCann, Robert J.; Schmuckenschläger, Michael
Prékopa–Leindler type inequalities on riemannian manifolds, Jacobi fields, and optimal transport. Annales de la faculté des sciences de Toulouse, Sér. 6, 15 no. 4 (2006), p. 613-635
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Nous étudions l’extension d’inégalités de type Prékopa-Leindler au cas d’une variété riemannienne $M$ équipée d’une mesure ayant une densité $e^{-V}$ où le potentiel $V$ et la courbure de Ricci vérifient $\operatorname{Hess}_x V + \operatorname{Ric}_x \ge \lambda \, I\ (\forall x\in M)$, pour un certain $\lambda \in \mathbb{R}$. Nous ferons appel, comme dans notre travail précédent [14], au transport optimal de mesure. Mais nous exploiterons plus encore son lien avec les champs de Jacobi, ce qui permettra de ramener la discussion à l’étude du déterminant d’une matrice de champs de Jacobi. Nous présentons également d’autres applications de la méthode, en particulier aux inégalités de Sobolev logarithmiques (critère de Bakry-Emery) et à l’étude de la convexité de déplacement de la fonctionnelle entropie.
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