On Kac's chaos and related problems

被引:77
作者
Hauray, Maxime [1 ]
Mischler, Stephane [2 ,3 ]
机构
[1] Univ Aix Marseille, CNRS, Cent Marseille, IMM UMR 7373, F-13453 Marseille, France
[2] Univ Paris 09, F-75775 Paris 16, France
[3] IUF, CEREMADE, UMR CNRS 7534, F-75775 Paris 16, France
关键词
Kac's chaos; Monge Kantorovich Wasserstein distance; Entropy chaos; Fisher information chaos; CLT with optimal rate; Probability measures mixtures; De Finetti; Hewitt and Savage theorem; Mean-field limit; Quantitative chaos; Qualitative chaos; 2-DIMENSIONAL EULER EQUATIONS; CENTRAL-LIMIT-THEOREM; STATISTICAL-MECHANICS; FISHER INFORMATION; STATIONARY FLOWS; ENTROPY; APPROXIMATION; INEQUALITIES; EQUILIBRIUM; PROPAGATION;
D O I
10.1016/j.jfa.2014.02.030
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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 -> infinity. 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. (C) 2014 Elsevier Inc. All rights reserved.
引用
收藏
页码:6055 / 6157
页数:103
相关论文
共 76 条
  • [1] ON OPTIMAL MATCHINGS
    AJTAI, M
    KOMLOS, J
    TUSNADY, G
    [J]. COMBINATORICA, 1984, 4 (04) : 259 - 264
  • [2] [Anonymous], HAL00609453 2
  • [3] [Anonymous], ARXIV12051241
  • [4] [Anonymous], ARXIV11043994V1
  • [5] [Anonymous], 1967, Commun. Math. Phys.
  • [6] [Anonymous], 1983, LECT NOTES CONTROL I
  • [7] [Anonymous], COURS C FRANC
  • [8] [Anonymous], 1933, Giorn. Ist. Ital. Attuari
  • [9] [Anonymous], CHAOS PROPERTIES BOL
  • [10] [Anonymous], ARXIV11032734V2