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 $\alpha$-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 $\alpha \alpha$. For this case, we provide sufficient conditions for the finiteness of the Kantorovich-Wasserstein distance.
Highlights
This paper is concerned with the study of the speed of convergence to equilibrium −with respect to Wasserstein distances − of the solution of the one–dimensional kinetic equation∂tμt + μt = Q+(μt, μt) μ0 = μ0. (1.1)The solution μt = μt(·) is a time-dependent probability measure on B(R), the Borel σfield of R
In the case of the Kac equation, that has the Gaussian distribution as steady state, rates of convergence with respect to Kolmogorov’s uniform metric, weighted χ-metrics of order p ≥ 2, Wasserstein metrics of order 1 and 2 and total variation distance have been proved
A probability measure μ0 belongs to the domain of normal attraction of a stable law of exponent α if for any sequence of i.i.d. real-valued random variables (Xn)n≥1 with common distribution μ0, there exists a sequence of real numbersn≥1 such that the law of n−1/α n i=1
Summary
This paper is concerned with the study of the speed of convergence to equilibrium −. [20]) conditions for the relaxation to the steady state, an important problem is to determine explicit rates of convergence to the equilibrium with respect to suitable probability metrics. In the case of the Kac equation, that has the Gaussian distribution as steady state, rates of convergence with respect to Kolmogorov’s uniform metric, weighted χ-metrics of order p ≥ 2, Wasserstein metrics of order 1 and 2 and total variation distance have been proved. The rest of the paper is organized as follows: Section 2 contains a brief summary of some known results on the relaxation to equilibrium for the solution of equation (1.1)(1.2).
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