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Article Dans Une Revue Journal of Statistical Physics Année : 2016

Uniform Contractivity in Wasserstein Metric for the Original 1D Kac’s Model

Maxime Hauray

Résumé

We study here a very popular 1D jump model introduced by Kac: it consists of $N$ velocities encountering random binary collisions at which they randomly exchange energy. We show the uniform (in $N$) exponential contractivity of the dynamics in a non-standard Monge-Kantorovich-Wasserstein: precisely the MKW metric of order 2 on the energy. The result is optimal in the sense that for each $N$, the contractivity constant is equal to the $L^2$ spectral gap of the generator associated to Kac's dynamic. As a corollary, we get an uniform but non optimal contractivity in the MKW metric of order $4$. We use a simple coupling that works better that the parallel one. The estimates are simple and new (to the best of our knowledge).
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Dates et versions

hal-01256859 , version 1 (06-10-2023)

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Maxime Hauray. Uniform Contractivity in Wasserstein Metric for the Original 1D Kac’s Model. Journal of Statistical Physics, 2016, 162 (6), pp.1566 - 1570. ⟨10.1007/s10955-016-1476-1⟩. ⟨hal-01256859⟩
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