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October 2012 Self-similar solutions in one-dimensional kinetic models: A probabilistic view
Federico Bassetti, Lucia Ladelli
Ann. Appl. Probab. 22(5): 1928-1961 (October 2012). DOI: 10.1214/11-AAP818


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.


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Federico Bassetti. Lucia Ladelli. "Self-similar solutions in one-dimensional kinetic models: A probabilistic view." Ann. Appl. Probab. 22 (5) 1928 - 1961, October 2012.


Published: October 2012
First available in Project Euclid: 12 October 2012

zbMATH: 1259.82087
MathSciNet: MR3025685
Digital Object Identifier: 10.1214/11-AAP818

Primary: 60F05
Secondary: 82C40

Keywords: central limit theorem , domain of normal attraction , Kac model , marked recursive $N$-ary random trees , self-similar solution , smoothing transformations , Stable law

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.22 • No. 5 • October 2012
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