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.

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 ×{1,2}× 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)= 0 t v(K(s))ds, où |v|1 pour k petit. Nous prouvons que le processus (NlnN) -1/2 Y(Nt) 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 ×{1,2}× 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 0 t v(K(s))ds, where |v|1 for small k. We prove that the rescaled process (NlnN) -1/2 Y(Nt) 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.

DOI : 10.1214/13-AIHP554
Classification : 82C44, 60K35, 60G70
Mots clés : anomalous thermal conductivity, kinetic limit, invariance principle
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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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