SELF-SIMILAR SOLUTIONS IN ONE-DIMENSIONAL KINETIC MODELS: A PROBABILISTIC VIEW

• 成果类型: Article
• 署名作者: Bassetti, Federico; Ladelli, Lucia
• 署名单位: University of Pavia; Polytechnic University of Milan
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP818
• 发表日期: 2012
• 页码: 1928-1961
• 关键词: CENTRAL-LIMIT-THEOREM; power-like tails; smoothing transformation; similar asymptotics; fixed-points; distributions; equation; gases

摘要

This paper deals with a class of Boltzmann equations on the real line, extensions of the well-known Kac caricature. A distinguishing feature of the corresponding equations is that therein, the collision gain operators are defined by N-linear smoothing transformations. These kind of problems have been studied, from an essentially analytic viewpoint, in a recent paper by Bobylev, Cercignani and Gamba [Comm. Math. Phys. 291 (2009) 599-644]. Instead, the present work rests exclusively on probabilistic methods, based on techniques pertaining to the classical central limit problem and to the so-called fixed-point equations for probability distributions. An advantage of resorting to methods from the probability theory is that the same results-relative to self-similar solutions-as those obtained by Bobylev, Cercignani and Gamba, are here deduced under weaker conditions. In particular, it is shown how convergence to a self-similar solution depends on the belonging of the initial datum to the domain of attraction of a specific stable distribution. Moreover, some results on the speed of convergence are given in terms of Kantorovich-Wasserstein and Zolotarev distances between probability measures.