From a kinetic equation to a diffusion under an anomalous scaling

Basile, Giada

doi:10.1214/13-AIHP554

Basile, Giada

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1301-1322.

Résumé
Abstract

Une équation de Boltzmann linéaire est interprétée comme équation de Fokker-Planck associée à la densité de probabilité d’un processus de Markov $(K (t), i (t), Y (t))$ sur $(𝕋^{2} \times {1, 2} \times ℝ^{2})$ , où $𝕋^{2}$ est le tore bidimensionnel. Le processus Markovien $(K (t), i (t))$ est ici un processus de sauts réversible avec des temps d’attente entre deux sauts à moyenne finie mais variance infinie. $Y (t)$ est une fonctionnelle additive de $K$ , définie par $Y (t) = \int_{0}^{t} v (K (s)) d s$ , où $| v | \sim 1$ pour $k$ petit. Nous prouvons que le processus ${(N ln N)}^{- 1 / 2} Y (N t)$ converge en distribution vers un mouvement brownien bidimensionnel. En conséquence, et moyennant un changement d’échelle approprié, la solution de l’équation de Boltzmann converge vers celle d’ une équation de diffusion.

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $(K (t), i (t), Y (t))$ on $(𝕋^{2} \times {1, 2} \times ℝ^{2})$ , where $𝕋^{2}$ is the two-dimensional torus. Here $(K (t), i (t))$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y (t)$ is an additive functional of $K$ , defined as $\int_{0}^{t} v (K (s)) d s$ , where $| v | \sim 1$ for small $k$ . We prove that the rescaled process ${(N ln N)}^{- 1 / 2} Y (N t)$ converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately rescaled solution of the Boltzmann equation converges to the solution of a diffusion equation.

MR Zbl

DOI : 10.1214/13-AIHP554

Classification : 82C44, 60K35, 60G70
Mots clés : anomalous thermal conductivity, kinetic limit, invariance principle

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     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1301--1322},
     publisher = {Gauthier-Villars},
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     url = {http://www.numdam.org/articles/10.1214/13-AIHP554/}
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Basile, Giada. From a kinetic equation to a diffusion under an anomalous scaling. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1301-1322. doi : 10.1214/13-AIHP554. http://www.numdam.org/articles/10.1214/13-AIHP554/

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