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

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.