The Bird and Nanbu systems are particle systems used to approximate the solution of the mollified Boltzmann equation. These systems have the propagation of chaos property. Following [6, 7, 8], we use coupling techniques to write a kind of expansion of the error in the propagation of chaos in terms of the number of particles. This expansion enables us to prove the a.s. convergence and the central-limit theorem for these systems. Notably, we obtain a central-limit theorem for the empirical measure of the system. As it is the case in [6, 7, 8], these results apply to the trajectories of particles on an interval .
Les systèmes de Bird et Nanbu sont des systèmes de particules en interaction approchant la solution de l’équation de Boltzmann mollifiée. Ces systèmes vérifient la propagation du chaos. Dans l’esprit de [6, 7, 8], nous utilisons des techniques de couplage pour écrire un développement asymptotique dans la propagation du chaos, en terme du nombre de particules. Ce développement nous permet de démontrer la convergence p.s. de ces systèmes, ainsi qu’un théorème central-limite. Ce théorème central-limite s’applique à la mesure empirique du système. Comme dans [6, 7, 8], ces résultats s’appliquent aux trajectoires des particules sur un intervalle .
@article{AFST_2016_6_25_4_829_0,
author = {Sylvain Rubenthaler},
title = {Central limit theorem through expansion of the propagation of chaos for {Bird} and {Nanbu} systems},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {829--873},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 25},
number = {4},
year = {2016},
doi = {10.5802/afst.1512},
zbl = {1355.82041},
mrnumber = {3564128},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1512/}
}
TY - JOUR AU - Sylvain Rubenthaler TI - Central limit theorem through expansion of the propagation of chaos for Bird and Nanbu systems JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2016 SP - 829 EP - 873 VL - 25 IS - 4 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1512/ DO - 10.5802/afst.1512 LA - en ID - AFST_2016_6_25_4_829_0 ER -
%0 Journal Article %A Sylvain Rubenthaler %T Central limit theorem through expansion of the propagation of chaos for Bird and Nanbu systems %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2016 %P 829-873 %V 25 %N 4 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1512/ %R 10.5802/afst.1512 %G en %F AFST_2016_6_25_4_829_0
Sylvain Rubenthaler. Central limit theorem through expansion of the propagation of chaos for Bird and Nanbu systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 4, pp. 829-873. doi : 10.5802/afst.1512. https://afst.centre-mersenne.org/articles/10.5802/afst.1512/
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