Quantitative propagation of chaos for generalized Kac particle systems

TL;DR

This paper establishes explicit, time-dependent convergence rates for particle systems with binary interactions, including models from kinetic theory and economics, using novel coupling techniques and empirical measure estimates.

Contribution

It introduces a new coupling method and provides explicit convergence rates for generalized Kac particle systems, extending previous results to more complex models.

Findings

Linear convergence rates in Wasserstein distance over time

Dependence of convergence on the number of particles is polynomial

Applicable to models in kinetic theory and economic dynamics

Abstract

We study a class of one-dimensional particle systems with true (Bird type) binary interactions, which includes Kac's model of the Boltzmann equation and nonlinear equations for the evolution of wealth distribution arising in kinetic economic models. We obtain explicit rates of convergence for the Wasserstein distance between the law of the particles and their limiting law, which are linear in time and depend in a mild polynomial manner on the number of particles. The proof is based on a novel coupling between the particle system and a suitable system of nonindependent nonlinear processes, as well as on recent sharp estimates for empirical measures.