@article{EJP2054,
author = {Federico Bassetti and Eleonora Perversi},
title = {Speed of convergence to equilibrium in Wasserstein metrics for Kac-like kinetic equations},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {18},
year = {2013},
keywords = {Boltzmann-like equations, Kac caricature, smoothing transformation, stable laws, rate of convergence to equilibrium, Wasserstein distances},
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 distancesof order $p>\alpha$, under the natural assumption that the distancebetween 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 distributionand 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.
},
pages = {no. 6, 1-35},
issn = {1083-6489},
doi = {10.1214/EJP.v18-2054},
url = {http://ejp.ejpecp.org/article/view/2054}}