[Sur les systèmes de particules en interaction singulière répulsive en dimension : log-gaz et gaz de Riesz]
In this article, we prove the first quantitative uniform in time propagation of chaos for a class of systems of particles in singular repulsive interaction in dimension one that contains the Dyson Brownian motion. We start by establishing existence and uniqueness for the Riesz gases, before proving propagation of chaos with an original approach to the problem, namely coupling with a Cauchy sequence type argument. We also give a general argument to turn a result of weak propagation of chaos into a strong and uniform in time result using the long time behavior and some bounds on moments, in particular enabling us to get a uniform in time version of the result of Cépa-Lépingle [CL97].
Dans cet article, nous prouvons le premier résultat de propagation du chaos quantitative uniforme en temps pour une classe de systèmes de particules en interaction singulière répulsive en dimension qui contient le mouvement brownien de Dyson. Nous commençons par établir l’existence et l’unicité des gaz de Riesz, avant de prouver la propagation du chaos par une approche originale du problème, à savoir un couplage avec un argument de type suite de Cauchy. Nous donnons également un argument général pour transformer un résultat faible de propagation du chaos en un résultat fort et uniforme en temps en utilisant le comportement en temps long et certaines bornes sur les moments, ce qui nous permet en particulier d’obtenir une version uniforme en temps du résultat de Cépa-Lépingle [CL97].
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Keywords: Propagation of chaos, long-time behavior, Riesz gas, Dyson Brownian motion, stochastic calculus
Mots-clés : Propagation du chaos, comportement en temps long, gaz de Riesz, mouvement brownien de Dyson, calcul stochastique
Guillin, Arnaud 1 ; Le Bris, Pierre 2 ; Monmarché, Pierre 2
CC-BY 4.0
@article{JEP_2023__10__867_0,
author = {Guillin, Arnaud and Le Bris, Pierre and Monmarch\'e, Pierre},
title = {On systems of particles in singular repulsive interaction in dimension one: log and {Riesz} gas},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {867--916},
year = {2023},
publisher = {Ecole polytechnique},
volume = {10},
doi = {10.5802/jep.235},
language = {en},
url = {https://www.numdam.org/articles/10.5802/jep.235/}
}
TY - JOUR AU - Guillin, Arnaud AU - Le Bris, Pierre AU - Monmarché, Pierre TI - On systems of particles in singular repulsive interaction in dimension one: log and Riesz gas JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 867 EP - 916 VL - 10 PB - Ecole polytechnique UR - https://www.numdam.org/articles/10.5802/jep.235/ DO - 10.5802/jep.235 LA - en ID - JEP_2023__10__867_0 ER -
%0 Journal Article %A Guillin, Arnaud %A Le Bris, Pierre %A Monmarché, Pierre %T On systems of particles in singular repulsive interaction in dimension one: log and Riesz gas %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 867-916 %V 10 %I Ecole polytechnique %U https://www.numdam.org/articles/10.5802/jep.235/ %R 10.5802/jep.235 %G en %F JEP_2023__10__867_0
Guillin, Arnaud; Le Bris, Pierre; Monmarché, Pierre. On systems of particles in singular repulsive interaction in dimension one: log and Riesz gas. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 867-916. doi: 10.5802/jep.235
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