Central limit theorem for a class of one-dimensional kinetic equations

Abstract

We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation's solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law ν α , then the limit is a scale mixture of ν α . Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.

Self-similar solutions of kinetic-type equations: The boundary case

“…The solution to the equation (1.1) can be derived analytically in terms of the Wild series [33], see also Kielek [26]. However, based on McKean's [29] ideas, Bassetti, Ladelli and Matthes [4,5] expressed the solution in a convenient probabilistic way. Ealier results on probabilistic representation can be found in [16,20,21].…”

“…The main goal of our paper is to present a class of solutions to (1.1) which, after rescaling as in (1.2), converge to a degenerate limit, yet it is possible to find a different normalization ensuring a nondegenerate limit possessing some self-similarity properties. To achieve our aims we propose a refinement of the probabilistic construction of the solution φ presented in [5] and express φ via a continuous-time branching random walk. Firstly, we state assumptions concerning the initial condition φ 0 .…”

Inequality and risk aversion in economies open to altruistic attitudes

“…As recalled in Section 2.1 in Bassetti et al (2011), the Fourier-Stieltjes transform of the solution µ t to (3) can be expressed aŝ…”

Wealth distribution in presence of debts: A Fokker–Planck description

“…All the previous results describe a society in which all agents have initially a non negative wealth, and do not consider the unpleasant but realistic possibility that part of the agents would have debts, clearly expressed by a negative wealth. Recent results on one-dimensional kinetic models [3,4] showed however that there are no mathematical obstacles in considering the Boltzmann-type equation introduced in [16] with initial values supported on the whole real line.…”

“…Following the idea of [3,4], we will study in this paper the initial value problem for the Fokker-Planck equation (1.2) posed on the whole real line R, by assuming that the initial datum satisfies condition (1.1), that is by assuming that part of the agents of the society could initially have debts, while the initial (conserved) mean wealth is positive. As we shall see, also in this situation, the positivity of the mean wealth will be enough to drive the solution towards the (unique) equilibrium density, still given by (1.3).…”