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Uniform propagation of chaos for Kac's 1D particle system (1606.06330v2)

Published 20 Jun 2016 in math.PR, math-ph, and math.MP

Abstract: In this paper we study Kac's 1D particle system, consisting of the velocities of $N$ particles colliding at constant rate and randomly exchanging energies. We prove uniform (in time) propagation of chaos in Wasserstein distance with explicit polynomial rates in $N$, for both the squared (i.e., the energy) and non-squared particle system. These rates are of order $N{-1/3}$ (almost, in the non-squared case), assuming that the initial distribution of the limit nonlinear equation has finite moments of sufficiently high order ($4+\epsilon$ is enough when using the 2-Wasserstein distance). The proof relies on a convenient parametrization of the collision recently introduced by Hauray, as well as on a coupling technique developed by Cortez and Fontbona.

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