Self-similar solutions in one-dimensional kinetic models: A probabilistic view

Abstract

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.

Published in at http://dx.doi.org/10.1214/11-AAP818 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails
• Self-similarity and power-like tails in nonconservative kinetic models
• The functional equation of the smoothing transform
• Self-Similarity in Random Collision Processes
• Central limit theorem for the solution of the Kac equation
• Kinetic models with randomly perturbed binary collisions

Cited by in corpus (5)

• Heavy tailed solutions of multivariate smoothing transforms
• Precise tail asymptotics of fixed points of the smoothing transform with general weights
• Characterization of weak convergence of probability-valued solutions of general one-dimensional kinetic equations
• Probabilistic View of Explosion in an Inelastic Kac Model
• Solutions of kinetic-type equations with perturbed collisions