Stochastic proof of the sharp symmetrized Talagrand inequality

Thomas A. Courtade; Max Fathi; Dan Mikulincer

doi:10.5802/crmath.681

Research article - Probability theory

Stochastic proof of the sharp symmetrized Talagrand inequality

Thomas A. Courtade ¹ ; Max Fathi ^2,^3,⁴; Dan Mikulincer ⁵

¹ Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720 USA
² Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions and Laboratoire de Probabilités, Statistique et Modélisation, F-75013 Paris, France
³ DMA, École normale supérieure, Université PSL, CNRS, 75005 Paris, France
⁴ Institut Universitaire de France
⁵ Department of Mathematics, MIT, Cambridge, MA 02139 USA

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1779-1784.

Abstract
Résumé

A new proof of the sharp symmetrized form of Talagrand’s transport-entropy inequality is given. Compared to stochastic proofs of other Gaussian functional inequalities, the new idea here is a certain coupling induced by time-reversed martingale representations.

Nous donnons une nouvelle preuve de la version symétrisée de l’inégalité de transport-entropie de Talagrand avec constante optimale. En comparaison avec d’autres preuves stochastiques d’inégalités fonctionnelles gaussiennes, l’élément nouveau ici est l’utilisation d’un couplage induit par un retournement du temps sur des représentations de martingales.

Received: 2024-07-12
Revised: 2024-08-22
Accepted: 2024-08-23
Published online: 2024-11-25

DOI: 10.5802/crmath.681

Classification: 60H30, 39B62, 52A40
Keywords: Transport inequalities, Gaussian inequalities, Blaschke–Santaló inequality, Martingale representations
Mots-clés : Inégalités de transport, inégalités Gaussiennes, inégalité de Blaschke–Santaló, représentations de martingales

Author's affiliations:

Thomas A. Courtade ¹; Max Fathi ^{2, 3, 4}; Dan Mikulincer ⁵

¹ Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720 USA
² Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions and Laboratoire de Probabilités, Statistique et Modélisation, F-75013 Paris, France
³ DMA, École normale supérieure, Université PSL, CNRS, 75005 Paris, France
⁴ Institut Universitaire de France
⁵ Department of Mathematics, MIT, Cambridge, MA 02139 USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Thomas A. Courtade and Max Fathi and Dan Mikulincer},
     title = {Stochastic proof of the sharp symmetrized {Talagrand} inequality},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1779--1784},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.681},
     language = {en},
}

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Thomas A. Courtade; Max Fathi; Dan Mikulincer. Stochastic proof of the sharp symmetrized Talagrand inequality. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1779-1784. doi : 10.5802/crmath.681. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.681/

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