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

TL;DR

This paper uses probabilistic methods to analyze self-similar solutions of one-dimensional Boltzmann equations, extending previous analytic results by establishing convergence conditions and rates based on stable distribution domains.

Contribution

It introduces a probabilistic approach to study self-similar solutions in kinetic models, relaxing conditions and providing convergence speed estimates.

Findings

Convergence to self-similar solutions depends on initial data being in a stable distribution domain.

Weaker conditions for convergence are established compared to previous analytic methods.

Speed of convergence is quantified using Kantorovich-Wasserstein and Zolotarev distances.

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.