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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
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