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Modulated logarithmic Sobolev inequalities and generation of chaos
Matthew Rosenzweig1 ; Sylvia Serfaty2
1 Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA, USA
2 Courant Institute of Mathematical Sciences, New York University, New York City, NY, USA
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 1, pp. 107-134.

We consider mean-field limits for overdamped Langevin dynamics of $N$ particles with possibly singular interactions. It has been shown that a modulated free energy method can be used to prove the mean-field convergence or propagation of chaos for a certain class of interactions, including Riesz kernels. We show here that generation of chaos, i.e. exponential in time convergence to a tensorized (or iid) state starting from a nontensorized one, can be deduced from the modulated free energy method provided a uniform-in-$N$ “modulated logarithmic Sobolev inequality” holds. Proving such an inequality is a question of independent interest, which is generally difficult. As an illustration, we show that uniform modulated logarithmic Sobolev inequalities can be proven for a class of situations in one dimension.

On considère la limite de champ moyen pour la dynamique de Langevin suramortie de $N$ particules en interaction (possiblement) singulière. Il a été montré qu’on peut utiliser une méthode d’énergie libre modulée pour traiter une certaine classe d’interactions qui inclut les potentiels de Riesz. Nous montrons ici que l’on peut déduire de la méthode d’énergie libre modulée la génération du chaos, c.à.d. la convergence exponentielle en temps vers un état tensorisé partant d’une situation initiale non tensorisée, à condition qu’une « inégalité de Sobolev logarithmique modulée » uniforme en $N$ soit vraie. Prouver une telle inégalité est une question indépendante qui est en général difficile. Comme illustration, nous montrons que des inégalités de Sobolev logarithmiques modulées peuvent être prouvées pour une classe de problèmes unidimensionnels.

Reçu le :
Accepté le :
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DOI : 10.5802/afst.1807

Matthew Rosenzweig 1 ; Sylvia Serfaty 2

1 Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA, USA
2 Courant Institute of Mathematical Sciences, New York University, New York City, NY, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Matthew Rosenzweig; Sylvia Serfaty. Modulated logarithmic Sobolev inequalities and generation of chaos. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 1, pp. 107-134. doi : 10.5802/afst.1807. https://afst.centre-mersenne.org/articles/10.5802/afst.1807/

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