Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Cordero-Erausquin, Dario; McCann, Robert J.; Schmuckenschläger, Michael

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 4, pp. 613-635.

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 ${Hess}_{x}V+{Ric}_x\ge \lambda I(\forall x\in M)$, pour un certain $\lambda \in ℝ$. 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.

We investigate Prékopa-Leindler type inequalities on a Riemannian manifold $M$ equipped with a measure with density $e^{-V}$ where the potential $V$ and the Ricci curvature satisfy ${Hess}_{x}V+{Ric}_x\ge \lambda I$ for all $x\in M$, with some $\lambda \in ℝ$. As in our earlier work [14], the argument uses optimal mass transport on $M$, but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to logarithmic Sobolev inequalities (the Bakry-Emery criterion will be recovered) and to transport inequalities. A study of the displacement convexity of the entropy functional completes the exposition.

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