This note is based on a talk given at the Séminaire Laurent Schwartz in February 2024. The goal is to review a recent joint work with Arnaud Guillin (Université Clermont-Auvergne) and Pierre Monmarché (Sorbonne Université) published in Journal de l’École polytechnique - Mathématiques (2023) [GLM23].
Consider a one dimensional -particle system in singular repulsive mean field interaction. The main motivating example is the (generalized) Dyson Brownian motion which holds importance in Random Matrix Theory. We wish to show the convergence, as goes to infinity, of the empirical distribution of the system towards the solution of a non linear PDE.
We describe a method that relies only on the well posedness of the system of particles and which provides a quantitative (and in some cases uniform in time) result. We make full use of the fact that in dimension one the particles will stay ordered, and that as a consequence the interaction we consider will be convex. Using a coupling method, we prove that by taking any independent sequence of empirical measures, it is a Cauchy sequence. Then, independence ensures the fact that the limit is an almost surely constant random variable which we then identify. This method requires in particular no study of the non linear limit.
@article{SLSEDP_2023-2024____A5_0, author = {Pierre Le Bris}, title = {Uniform in time mean field limits for {1D} {Riesz} gases}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:11}, pages = {1--11}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2023-2024}, doi = {10.5802/slsedp.168}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.168/} }
TY - JOUR AU - Pierre Le Bris TI - Uniform in time mean field limits for 1D Riesz gases JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:11 PY - 2023-2024 SP - 1 EP - 11 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.168/ DO - 10.5802/slsedp.168 LA - en ID - SLSEDP_2023-2024____A5_0 ER -
%0 Journal Article %A Pierre Le Bris %T Uniform in time mean field limits for 1D Riesz gases %J Séminaire Laurent Schwartz — EDP et applications %Z talk:11 %D 2023-2024 %P 1-11 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.168/ %R 10.5802/slsedp.168 %G en %F SLSEDP_2023-2024____A5_0
Pierre Le Bris. Uniform in time mean field limits for 1D Riesz gases. Séminaire Laurent Schwartz — EDP et applications (2023-2024), Exposé no. 11, 11 p. doi : 10.5802/slsedp.168. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.168/
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