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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