## Speed of convergence to equilibrium in Wasserstein metrics for Kac-like kinetic equations.(English)Zbl 1408.60005

Summary: 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.

### MSC:

 60B10 Convergence of probability measures 82C40 Kinetic theory of gases in time-dependent statistical mechanics 60E07 Infinitely divisible distributions; stable distributions 60F05 Central limit and other weak theorems
Full Text: