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

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 α-stable law, the solution of the equation converges weakly to a suitable scale mixture of centered α-stable laws. In this paper we present explicit exponential rates for the convergence to equilibrium in Kantorovich-Wasserstein distances of order p > α, under the natural assumption that the distance between the initial datum and the limit distribution is finite. For α = 2 this assumption reduces to the finiteness of the absolute moment of order p of the initial datum. On the contrary, when α < 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 > α. For this case, we provide sufficient conditions for the finiteness of the Kantorovich-Wasserstein distance.Date: November 12, 2018. Key words and phrases. Boltzmann-like equations and Kac caricature and stable laws and rate of convergence to equilibrium and Wasserstein distances .

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