Wasserstein convergence rates for stochastic particle approximation of Boltzmann models

Abstract

We establish quantitative convergence rates for stochastic particle approximation based on Nanbu-type Monte Carlo schemes applied to a broad class of collisional kinetic models. Using coupling techniques and stability estimates in the Wasserstein-1 (Kantorovich-Rubinstein) metric, we derive sharp error bounds that reflect the nonlinear interaction structure of the models. Our framework includes classical Nanbu Monte Carlo method and more recent developments as Time Relaxed Monte Carlo methods. The results bridge the gap between probabilistic particle approximations and deterministic numerical error analysis, and provide a unified perspective for the convergence theory of Monte Carlo methods for Boltzmann-type equations. As a by-product, we also obtain existence and uniqueness of solutions to a large class of Boltzmann-type equations.