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

摘要

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 $lpha$-stable law, the solution of the equation converges weakly to a suitable scale mixture of centered $lpha$-stable laws. In this paper we present explicit exponential rates for the convergence to equilibrium in Kantorovich-Wasserstein distancesof order $p>lpha$, under the natural assumption that the distancebetween the initial datum and the limit distribution is finite. For $lpha=2$ this assumption reduces to the finiteness of the absolute moment of order $p$ of the initial datum. On the contrary, when $lphalpha$. For this case, we provide sufficient conditions for the finiteness of the Kantorovich-Wasserstein distance.