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].
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].
CC-BY 4.0
@article{JEP_2023__10__867_0,
author = {Arnaud Guillin and Pierre Le Bris and Pierre Monmarch\'e},
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},
publisher = {\'Ecole polytechnique},
volume = {10},
year = {2023},
doi = {10.5802/jep.235},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.235/}
}
TY - JOUR AU - Arnaud Guillin AU - Pierre Le Bris AU - Pierre Monmarché 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 - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.235/ DO - 10.5802/jep.235 LA - en ID - JEP_2023__10__867_0 ER -
%0 Journal Article %A Arnaud Guillin %A Pierre Le Bris %A Pierre Monmarché %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 École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.235/ %R 10.5802/jep.235 %G en %F JEP_2023__10__867_0
Arnaud Guillin; Pierre Le Bris; Pierre Monmarché. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.235/
[AGZ10] - An introduction to random matrices, Cambridge Studies in Advanced Math., vol. 118, Cambridge University Press, Cambridge, 2010
[BDLL22] - “A spectral dominance approach to large random matrices”, J. Math. Pures Appl. (9) 164 (2022), p. 27-56 | DOI | MR | Zbl
[BJW20] - “Modulated free energy and mean field limit”, in Séminaire Laurent Schwartz—EDP et applications, vol. 2019-2020, Éditions de l’École polytechnique, Palaiseau, 2020, 22 p. | Numdam | Zbl
[BO19] - “Propagation of chaos for a class of first order models with singular mean field interactions”, SIAM J. Math. Anal. 51 (2019) no. 1, p. 159-196 | DOI | MR | Zbl
[Bol08] - “Separability and completeness for the Wasserstein distance”, in Séminaire de probabilités XLI, Lect. Notes in Math., vol. 1934, Springer, Berlin, 2008, p. 371-377 | DOI | MR | Zbl
[CGM08] - “Probabilistic approach for granular media equations in the non-uniformly convex case”, Probab. Theory Relat. Fields 140 (2008) no. 1-2, p. 19-40 | DOI | MR | Zbl
[Cha92] - “The Wigner semi-circle law and eigenvalues of matrix-valued diffusions”, Probab. Theory Relat. Fields 93 (1992) no. 2, p. 249-272 | DOI | MR | Zbl
[CL97] - “Diffusing particles with electrostatic repulsion”, Probab. Theory Relat. Fields 107 (1997) no. 4, p. 429-449 | DOI | MR | Zbl
[CL20] - “On Poincaré and logarithmic Sobolev inequalities for a class of singular Gibbs measures”, in Geometric aspects of functional analysis. Vol. I, Lect. Notes in Math., vol. 2256, Springer, Cham, 2020, p. 219-246 | DOI | Zbl
[CP18] - “Computational optimal transport”, Found. and Trends in Machine Learning 11 (2018) no. 5-6, p. 355-206 | DOI
[DEGZ20] - “An elementary approach to uniform in time propagation of chaos”, Proc. Amer. Math. Soc. 148 (2020) no. 12, p. 5387-5398 | DOI | MR | Zbl
[Dys62] - “A Brownian-motion model for the eigenvalues of a random matrix”, J. Math. Phys. 3 (1962), p. 1191-1198 | DOI | MR
[Ebe16] - “Reflection couplings and contraction rates for diffusions”, Probab. Theory Relat. Fields 166 (2016) no. 3-4, p. 851-886 | DOI | MR | Zbl
[EGZ19] - “Quantitative Harris-type theorems for diffusions and McKean-Vlasov processes”, Trans. Amer. Math. Soc. 371 (2019) no. 10, p. 7135-7173 | DOI | MR | Zbl
[HM19] - “Ergodicity and Lyapunov functions for Langevin dynamics with singular potentials”, Comm. Pure Appl. Math. 72 (2019) no. 10, p. 2231-2255 | DOI | MR | Zbl
[HS19] - “Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D”, Kinet. and Relat. Mod. 12 (2019) no. 2, p. 269-302 | DOI | MR | Zbl
[JW18] - “Quantitative estimates of propagation of chaos for stochastic systems with kernels”, Invent. Math. 214 (2018) no. 1, p. 523-591 | DOI | MR | Zbl
[Kac56] - “Foundations of kinetic theory”, in Proc. of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, University of California Press, Berkeley-Los Angeles, Calif., 1956, p. 171-197 | Zbl
[LLX20] - “On the law of large numbers for the empirical measure process of generalized Dyson Brownian motion”, J. Statist. Phys. 181 (2020) no. 4, p. 1277-1305 | DOI | MR | Zbl
[LM20] - “Geometric ergodicity of Langevin dynamics with Coulomb interactions”, Nonlinearity 33 (2020) no. 2, p. 675-699 | DOI | MR | Zbl
[McK67] - “Propagation of chaos for a class of non-linear parabolic equations”, in Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), Air Force Office Sci. Res., Arlington, Va., 1967, p. 41-57
[RS93] - “Interacting Brownian particles and the Wigner law”, Probab. Theory Relat. Fields 95 (1993) no. 4, p. 555-570 | DOI | MR | Zbl
[RS23] - “Global-in-time mean-field convergence for singular Riesz-type diffusive flows”, Ann. Probab. 33 (2023) no. 2, p. 754-798 | MR
[Ser18] - “Systems of points with Coulomb interactions”, in Proc. of the I.C.M—Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018, p. 935-977 | Zbl
[Ser20] - “Mean field limit for Coulomb-type flows”, Duke Math. J. 169 (2020) no. 15, p. 2887-2935 | DOI | MR | Zbl
[Szn91] - “Topics in propagation of chaos”, in École d’Été de Probabilités de Saint-Flour XIX—1989, Lect. Notes in Math., vol. 1464, Springer, Berlin, 1991, p. 165-251 | DOI | MR | Zbl
[Vil09] - Optimal transport : Old and new, Grundlehren Math. Wiss., vol. 338, Springer-Verlag, Berlin, 2009 | DOI | Numdam
[Wig55] - “Characteristic vectors of bordered matrices with infinite dimensions”, Ann. of Math. (2) 62 (1955), p. 548-564 | DOI | MR | Zbl
Cité par Sources :