On Kac's chaos and related problems

This paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by M. Kac [41] in his study of mean-field limit for systems of N   undistinguishable particles as N→∞N→∞. First, we quantitatively liken three usual measures of Kac's chaos, some involving all the N variables, others involving a finite fixed number of variables. Next, we define the notion of entropy chaos and Fisher information chaos in a similar way as defined by Carlen et al. [17]. We show that Fisher information chaos is stronger than entropy chaos, which in turn is stronger than Kac's chaos. We also establish that Kac's chaos plus Fisher information bound implies entropy chaos. We then extend our analysis to the framework of probability measures with support on the Kac's spheres, revisiting [17] and giving a possible answer to [17, Open problem 11]. Last, we consider the context of probability measures mixtures introduced by De Finetti, Hewitt and Savage. We define the (level 3) Fisher information for mixtures and prove that it is l.s.c. and affine, as that was done in [64] for the level 3 Boltzmann's entropy.