Speed of convergence to equilibrium in Wasserstein metrics for Kac-s like kinetic equations

TL;DR

This paper investigates the rate at which solutions to certain one-dimensional kinetic equations, extending Kac's model, converge to equilibrium in Wasserstein metrics, providing explicit exponential convergence rates under specific conditions.

Contribution

It offers explicit exponential convergence rates in Wasserstein distances for a class of kinetic equations, including cases with infinite moments, extending previous results on convergence to equilibrium.

Findings

Explicit exponential convergence rates in Wasserstein distances for -stable laws.

Conditions for finiteness of Wasserstein distance when moments are infinite.

Analysis of convergence behavior for <2 cases with infinite moments.

Abstract

This work deals with a class of one-dimensional measure-valued kinetic equations, which constitute extensions of the Kac caricature. It is known that if the initial datum belongs to the domain of normal attraction of an \alpha-stable law, the solution of the equation converges weakly to a suitable scale mixture of centered \alpha-stable laws. In this paper we present explicit exponential rates for the convergence to equilibrium in Kantorovich-Wasserstein distances of order p>\alpha, under the natural assumption that the distance between the initial datum and the limit distribution is finite. For \alpha=2 this assumption reduces to the finiteness of the absolute moment of order p of the initial datum. On the contrary, when \alpha<2, the situation is more problematic due to the fact that both the limit distribution and the initial datum have infinite absolute moment of any order p >\alpha. For this case, we provide sufficient conditions for the finiteness of the Kantorovich-Wasserstein distance.