On Kac's chaos and related problems

By M. Hauray, St'ephane Mischler. 2012

167 citations recorded by the source corpus.

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.

Subfields: Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/15cb52801c3b003262c67ec156f518f87580d7df

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

A review of the mean field limits for Vlasov equations (2014, background): “In general this only occurs in some asymptotic sense as N → ∞ but not exactly for any finite N , leading to the notion of chaotic sequences of initial data as introduced in [104] (see also [36, 88] Definition 3.”
The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, methodology): “es rigorously follows from the property of propagation of chaos. For a classical introduction to these topics, we refer to [75]. Later, this property has been studied and developed in kinetic theory, [54, 55, 44, 50, 65]. Let us introduce the notion of propagation of chaos. Let us consider ρN(t,x 1,”
Quantitative estimates of propagation of chaos for stochastic systems with W-1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d (2017, background): “dxk ≤ HN (ρN | ρ̄N ), (12) for which we refer to [50,69,70] where estimates quantifying the classical notion of propagation of chaos are thoroughly investigated.”
Kac’s program in kinetic theory (2011, methodology): “ quick look at Theorem 5.1 shows that they scale like ℓ2, therefore making impossible to choose ℓ∼ N. Step 2: Inﬁnite-dimensional Wasserstein chaos. We shall use here the following result obtained in [36], see also [56, Th´eor`eme 2.1]: for any f ∈ P(Rd) and sequence fN ∈ P sym(Rd) we have ∀1 ≤ ℓ≤ N”
Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions (2021, background): “Note that there is no normalization by the dimension in our definitions ofH andWp, in contrast to [32, 34, 48, 28].”
Mean Field Limit for Stochastic Particle Systems (2017, methodology): “4 in [41], without having to know or prove anything about the mean field limit.”
Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces (2015, methodology): “Such stronger notions of propagation of chaos have recently been more thoroughly investigated and some of the connections between them elucidated, we refer to [29, 38, 39] for example or to the survey [31].”