Pathwise Convergence of the Hard Spheres Kac Process
Abstract
We derive two estimates for the deviation of the -particle, hard-spheres Kac process from the corresponding Boltzmann equation, measured in expected Wasserstein distance. Particular care is paid to the long-time properties of our estimates, exploiting the stability properties of the limiting Boltzmann equation at the level of realisations of the interacting particle system. As a consequence, we obtain an estimate for the propagation of chaos, uniformly in time and with polynomial rates, as soon as the initial data has a moment, . Our approach is similar to Kac's proposal of relating the long-time behaviour of the particle system to that of the limit equation. Along the way, we prove a new estimate for the continuity of the Boltzmann flow measured in Wasserstein distance.
Cite
@article{arxiv.1801.05791,
title = {Pathwise Convergence of the Hard Spheres Kac Process},
author = {Daniel Heydecker},
journal= {arXiv preprint arXiv:1801.05791},
year = {2019}
}
Comments
Supersedes the previous version. A version of the main theorem has been added which only requires $k$ initial moments, for any $k>2$. The continuity of the Boltzmann flow has been strengthened, and is now displayed as a separate theorem