Abstract
We derive two estimates for the deviation of the $N$-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 $k$th moment, $k>2$. 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.
Citation
Daniel Heydecker. "Pathwise convergence of the hard spheres Kac process." Ann. Appl. Probab. 29 (5) 3062 - 3127, October 2019. https://doi.org/10.1214/19-AAP1475
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