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Pré-Publication, Document De Travail Année : 2012

ON KAC'S CHAOS AND RELATED PROBLEMS

Résumé

This paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by M. Kac \cite{Kac1956} in his study of mean-field limit for systems of $N$ undistinguishable particles as $N\to\infty$. First, we quantitatively liken three usual measures of {\it Kac's chaos}, some involving the all $N$ variables, other involving a finite fixed number of variables. The cornerstone of the proof is a new representation of the Monge-Kantorovich-Wasserstein (MKW) distance for symmetric $N$-particle probabilities in terms of the distance between the law of the associated empirical measures on the one hand, and a new estimate on some MKW distance on probability spaces endowed with a suitable Hilbert norm taking advantage of the associated good algebraic structure. Next, we define the notion of {\it entropy chaos } and {\it Fisher information chaos } in a similar way as defined by Carlen et al \cite{CCLLV}. We show that {\it Fisher information chaos } is stronger than {\it entropy chaos}, which in turn is stronger than {\it Kac's chaos}. More importantly, with the help of the HWI inequality of Otto-Villani, we establish a quantitative estimate between these quantities, which in particular asserts that {\it Kac's chaos} plus {\it Fisher information bound } implies {\it entropy chaos}. We then extend the above quantitative and qualitative results about chaos in two other frameworks. We first extend it to the framework of probabilities with support on the {\it Kac's spheres}, revisiting \cite{CCLLV} and giving a possible answer to \cite[Open problem 11]{CCLLV}. Additionally to the above mentioned tool, we use and prove an optimal rate local CLT in $L^\infty$ norm for distributions with finite 6-th moment and finite $L^p$ norm, $p>1$. Last, we investigate how our techniques can be used in the context of probabilities mixtures as introduced by De Finetti, Hewitt and Savage and general sequence of probability densities of $N$ undistinguishable particles as $N\to\infty$, without assuming chaos.
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Dates et versions

hal-00682782 , version 1 (26-03-2012)
hal-00682782 , version 2 (19-05-2012)
hal-00682782 , version 3 (04-09-2012)
hal-00682782 , version 4 (23-11-2012)
hal-00682782 , version 5 (29-03-2013)

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  • HAL Id : hal-00682782 , version 1

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Maxime Hauray, Stéphane Mischler. ON KAC'S CHAOS AND RELATED PROBLEMS. 2012. ⟨hal-00682782v1⟩
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