Central limit theorems for solutions of the Kac equation: speed of approach to equilibrium in weak metrics

Abstract

This paper is part of our efforts to show how direct application of probabilistic methods, pertaining to central limit general theory, can enlighten us about the convergence to equilibrium of the solutions of the Kac equation. Here, we consider convergence with respect to the following metrics: Kolmogorov’s uniform metric; 1 and 2 Gini’s dissimilarity indices (widely known as 1 and 2 Wasserstein metrics); χ-weighted metrics. Our main results provide new bounds, or improvements on already well-known ones, for the corresponding distances between the solution of the Kac equation and the limiting Gaussian (Maxwellian) distribution. The study is conducted both under the necessary assumption that initial data have finite energy, without assuming existence of moments of order greater than 2, and under the condition that the (2 + δ)-moment of the initial distribution is finite for some δ > 0.