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 probability measures 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 measures 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 the framework of probability measures 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, for some $p>1$. Last, we investigate how our techniques can be used without assuming chaos, in the context of probability measures mixtures introduced by De Finetti, Hewitt and Savage. In particular, we define the (level 3) Fisher information for mixtures and prove that it is l.s.c. and affine, as that was done in \cite{RR} for the level 3 Boltzmann's entropy.
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