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2013 Speed of convergence to equilibrium in Wasserstein metrics for Kac-like kinetic equations
Federico Bassetti, Eleonora Perversi
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Electron. J. Probab. 18: 1-35 (2013). DOI: 10.1214/EJP.v18-2054


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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Federico Bassetti. Eleonora Perversi. "Speed of convergence to equilibrium in Wasserstein metrics for Kac-like kinetic equations." Electron. J. Probab. 18 1 - 35, 2013.


Accepted: 11 January 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 06247175
MathSciNet: MR3024100
Digital Object Identifier: 10.1214/EJP.v18-2054

Primary: 60B10
Secondary: 60E07 , 60F05 , 82C40

Keywords: Boltzmann-like equations , Kac caricature , Rate of convergence to equilibrium , Smoothing transformation , Stable laws , Wasserstein distances

Vol.18 • 2013
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