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

Federico Bassetti (University of Pavia)
Eleonora Perversi (University of Pavia)


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.

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Pages: 1-35

Publication Date: January 11, 2013

DOI: 10.1214/EJP.v18-2054


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