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Uniform-in-time estimates on corrections to mean field for interacting Brownian particles

Armand Bernou, Mitia Duerinckx

TL;DR

This work delivers uniform-in-time, higher-order propagation of chaos for interacting Brownian particles under smooth mean-field interactions, covering both Langevin and overdamped dynamics. The authors develop Lions expansions on the space of probability measures, complemented by Glauber calculus and novel Lions graphs to capture cancellations in cumulant expansions, yielding sharp N-dependent bounds G^{m,N}_t = O(N^{1-m}) and Bogolyubov-type corrections. They establish a uniform-in-time quantitative central limit theorem via a Gaussian Dean–Kawasaki limit and uniform concentration of the empirical measure, with expansions of E[Φ(μ^N_t)] in powers of 1/N. A key technical advance is the ergodic control of the linearized mean-field equation in weighted Sobolev spaces W^{-k,q}(⟨z⟩^p) for arbitrarily small p>0 and q>1, enabling long-time stability and enabling the uniform estimates. Collectively, the results connect mean-field approximations with fluctuation theories and long-time relaxation, with potential implications for mean-field games and related stochastic interacting particle systems.

Abstract

We consider a system of classical Brownian particles interacting via a smooth long-range potential in the mean-field regime, and we analyze the propagation of chaos in form of sharp, uniform-in-time estimates on many-particle correlation functions. Our results cover both the kinetic Langevin setting and the corresponding overdamped Brownian dynamics. The approach is mainly based on so-called Lions expansions, which we combine with new diagrammatic tools to capture many-particle cancellations, as well as with fine ergodic estimates on the linearized mean-field equation, and with discrete stochastic calculus with respect to initial data. In the process, we derive some new ergodic estimates for the linearized Vlasov-Fokker-Planck kinetic equation that are of independent interest. Our analysis also leads to a uniform-in-time quantitative central limit theorem and to uniform-in-time concentration estimates for the empirical measure associated with the particle dynamics.

Uniform-in-time estimates on corrections to mean field for interacting Brownian particles

TL;DR

This work delivers uniform-in-time, higher-order propagation of chaos for interacting Brownian particles under smooth mean-field interactions, covering both Langevin and overdamped dynamics. The authors develop Lions expansions on the space of probability measures, complemented by Glauber calculus and novel Lions graphs to capture cancellations in cumulant expansions, yielding sharp N-dependent bounds G^{m,N}_t = O(N^{1-m}) and Bogolyubov-type corrections. They establish a uniform-in-time quantitative central limit theorem via a Gaussian Dean–Kawasaki limit and uniform concentration of the empirical measure, with expansions of E[Φ(μ^N_t)] in powers of 1/N. A key technical advance is the ergodic control of the linearized mean-field equation in weighted Sobolev spaces W^{-k,q}(⟨z⟩^p) for arbitrarily small p>0 and q>1, enabling long-time stability and enabling the uniform estimates. Collectively, the results connect mean-field approximations with fluctuation theories and long-time relaxation, with potential implications for mean-field games and related stochastic interacting particle systems.

Abstract

We consider a system of classical Brownian particles interacting via a smooth long-range potential in the mean-field regime, and we analyze the propagation of chaos in form of sharp, uniform-in-time estimates on many-particle correlation functions. Our results cover both the kinetic Langevin setting and the corresponding overdamped Brownian dynamics. The approach is mainly based on so-called Lions expansions, which we combine with new diagrammatic tools to capture many-particle cancellations, as well as with fine ergodic estimates on the linearized mean-field equation, and with discrete stochastic calculus with respect to initial data. In the process, we derive some new ergodic estimates for the linearized Vlasov-Fokker-Planck kinetic equation that are of independent interest. Our analysis also leads to a uniform-in-time quantitative central limit theorem and to uniform-in-time concentration estimates for the empirical measure associated with the particle dynamics.
Paper Structure (52 sections, 35 theorems, 473 equations)

This paper contains 52 sections, 35 theorems, 473 equations.

Table of Contents

  1. Introduction
  2. General overview
  3. Main results
  4. Extensions
  5. Periodic setting
  6. Non-quadratic confinement
  7. More general interactions
  8. Strategy and plan of the paper
  9. Notation
  10. Preliminary
  11. Lions calculus
  12. Linear derivative
  13. L-derivative
  14. Master equation formalism
  15. Expansions along the particle dynamics
  16. ...and 37 more sections

Key Result

Theorem 1.1

Consider the Langevin dynamics eq:Langevin and the associated correlation functions $\{G^{m,N}\}_{2\le m\le N}$ as defined in eq:def-cumGm, and assume that the initial law has stretched exponential moments eq:mom-mu0 for some $\theta>0$. For all $m\ge2$, there exist $\kappa_m>0$ (only depending on $

Theorems & Definitions (63)

  • Theorem 1.1: Uniform-in-time higher-order propagation of chaos
  • Remark 1.2
  • Corollary 1.3: Bogolyubov correction to mean field
  • Theorem 1.4: Uniform-in-time CLT
  • Remark 1.5: Convergence rate
  • Remark 1.6: Higher-order fluctuations
  • Theorem 1.7: Uniform-in-time concentration
  • Lemma 2.1: see Carmona_2018Chassagneux_2019
  • Corollary 2.2
  • proof
  • ...and 53 more