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Quantitative and qualitative Kac's chaos on the Boltzmann's sphere

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Date
2015
Link to item file
https://arxiv.org/abs/1205.1241v3
Dewey
Probabilités et mathématiques appliquées
Sujet
mean-field limit
Journal issue
Annales de l'I.H.P. Probabilités et Statistiques
Volume
51
Number
3
Publication date
2015
Article pages
993-1039
Publisher
Gauthier-Villars
DOI
http://dx.doi.org/10.1214/14-AIHP612
Forthcoming
oui
URI
https://basepub.dauphine.fr/handle/123456789/9248
Collections
  • CEREMADE : Publications
Metadata
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Author
Carrapatoso, Kleber
Type
Article accepté pour publication ou publié
Abstract (EN)
We investigate the construction of chaotic probability measures on the Boltzmann's sphere, which is the state space of the stochastic process of a many-particle system undergoing a dynamics preserving energy and momentum. Firstly, based on a version of the local Central Limit Theorem (or Berry-Essenn theorem), we construct a sequence of probabilities that is Kac chaotic and we prove a quantitative rate of convergence. Then, we investigate a stronger notion of chaos, namely entropic chaos introduced in \cite{CCLLV}, and we prove, with quantitative rate, that this same sequence is also entropically chaotic. Furthermore, we investigate more general class of probability measures on the Boltzmann's sphere. Using the HWI inequality we prove that a Kac chaotic probability with bounded Fisher's information is entropically chaotic and we give a quantitative rate. We also link different notions of chaos, proving that Fisher's information chaos, introduced in \cite{HaurayMischler}, is stronger than entropic chaos, which is stronger than Kac's chaos. We give a possible answer to \cite[Open Problem 11]{CCLLV} in the Boltzmann's sphere's framework. Finally, applying our previous results to the recent results on propagation of chaos for the Boltzmann equation \cite{MMchaos}, we prove a quantitative rate for the propagation of entropic chaos for the Boltzmann equation with Maxwellian molecules.

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