Otto calculus has established itself as a powerful tool for proving quantitative energy dissipation estimates and provides with an elegant geometric interpretation of certain functional inequalities such as the Logarithmic Sobolev inequality. However, the local versions of such inequalities, which can be proven by means of Bakry–Émery–Ledoux -calculus, have not yet been given an interpretation in terms of this Riemannian formalism. In this short note we close this gap by explaining how Otto calculus applied to the Schrödinger problem yields a variations interpretation of local logarithmic Sobolev inequalities that could possibly unveil a novel class of local inequalities.
Le calcul d’Otto est un outil puissant pour quantifier la dissipation d’énergie mais surtout il propose une interprétation géométrique simple de certaines inégalités fonctionnelles comme l’inégalité de Sobolev logarithmique. La version locale de ces inégalités, dont la démonstration repose sur le -calcul développé par Bakry, Émery et Ledoux, n’avait pas encore une interprétation géométrique à la Otto. Dans cette courte note, nous comblons cette lacune et montrons comment le calcul d’Otto appliqué au problème de Schrödinger permet une interprétation variationnelle des inégalités de type Sobolev logarithmique locale, ce qui pourrait être un espoir pour l’élaboration de nouvelles d’inégalités locales.
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Gauthier Clerc 1; Giovanni Conforti 2; Ivan Gentil 1
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@article{AFST_2023_6_32_5_823_0,
author = {Gauthier Clerc and Giovanni Conforti and Ivan Gentil},
title = {On the variational interpretation of local logarithmic {Sobolev} inequalities},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {823--837},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 32},
number = {5},
year = {2023},
doi = {10.5802/afst.1754},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1754/}
}
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Gauthier Clerc; Giovanni Conforti; Ivan Gentil. On the variational interpretation of local logarithmic Sobolev inequalities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 5, pp. 823-837. doi : 10.5802/afst.1754. https://afst.centre-mersenne.org/articles/10.5802/afst.1754/
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