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

Federico Bassetti, Eleonora Perversi
2013 Electronic Journal of Probability  
E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. 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
more » ... gence 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. Speed of convergence in Wasserstein metrics metrics have been derived. For the solutions of the general model (1.1)-(1.2) less is known. Some results for the Wasserstein distances, of order p ≤ 2 have been proved in [2, 3]. The aim of this article is to prove new exponential bounds for the speed of approach to equilibrium for the solution of (1.1)-(1.2) with respect to Wasserstein metrics of any order. Our main results from Theorems 3.4, 3.5 and 3.14 can be summarized as follows: Assume that L and R are positive random variables such that P{L > 0} + P{R > 0} > 1, (1.3) holds with α ∈ (0, 1) ∪ (1, 2] and E[L p + R p ] < 1 for some p > α. Ifμ 0 belongs to the domain of normal attraction of an α-stable law (μ 0 being centered if α > 1) and the Wasserstein distance d p (μ 0 , µ ∞ ) is finite, then d p (µ t , µ ∞ ) ≤ Cμ 0 ,p e −Kα,pt for suitable positive constants Cμ 0 ,p and K α,p . A similar result holds for α = 1, see Theorem 3.5. The constant K α,p , that will be explicitly computed for α ≤ 2, depends only on the law of (L, R), while Cμ 0,p depends also onμ 0 and is finite if d p (μ 0 , µ ∞ ) < +∞. It is worth noticing that, if α < 2, the assumption d p (μ 0 , µ ∞ ) < +∞ is a non-trivial requirement, since, with the exception of some degenerate case, one has that R |x| pμ 0 (dx) = +∞ and R |x| p µ ∞ (dx) = +∞ for every p > α. For this reason, sufficient conditions for the finiteness of d p (μ 0 , µ ∞ ) will be Remark 2.4. It is worth noticing that the steady states µ ∞ described in Theorems 2.2-2.3 are the unique possible fixed points of Q + . See Theorems 2.1 and 2.2 in [1]. Necessary conditions for the convergence of µ t to a steady state µ ∞ are investigated in [28] .
doi:10.1214/ejp.v18-2054 fatcat:x433rbgmrfhfdijldeecqyk2dy