On Kac's Chaos And Related Problems
Maxime Hauray (LATP), St\'ephane Mischler (CEREMADE)

TL;DR
This paper investigates quantitative and qualitative aspects of Kac's chaos, introducing entropy and Fisher information chaos, establishing their relationships, and extending results to Kac's spheres and mixture models.
Contribution
It defines and compares entropy and Fisher information chaos, establishing their hierarchy and relationships with Kac's chaos, and extends these concepts to Kac's spheres and mixture measures.
Findings
Fisher information chaos is stronger than entropy chaos, which is stronger than Kac's chaos.
Quantitative estimates relate chaos measures via the HWI inequality.
Optimal rate local CLT in $L^ Infty$ norm for distributions with finite moments.
Abstract
This paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by M. Kac in his study of mean-field limit for systems of undistinguishable particles. First, we quantitatively liken three usual measures of Kac's chaos, some involving the all variables, other 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 (KRM 2010). We show that Fisher information chaos is stronger than entropy chaos, which in turn is stronger than 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 Kac's chaos plus Fisher information bound implies entropy chaos. We then extend the above quantitative and…
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Taxonomy
TopicsMathematics and Applications
