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Correlation estimates for Brownian particles with singular interactions

Mitia Duerinckx, Pierre-Emmanuel Jabin

TL;DR

The paper develops a linearized correlation framework for second-order Langevin particle systems with singular, potentially unbounded interactions in the mean-field regime. By introducing linear correlations H_{N,m} and a tractable BBGKY-type hierarchy, it obtains robust L^2-type bounds and weak convergence results for G_{N,m} under square-integrable kernels, and extends the analysis to the overdamped case. The authors establish a Bogolyubov correction with propagation of chaos at rate N^{−1}, characterize a Gaussian CLT for empirical measure fluctuations via a linearized Dean–Kawasaki SPDE, and, in the bounded-forces regime, derive globally valid, sharper estimates with optimal scaling. These results significantly extend correlation control to singular interaction regimes and provide a rigorous framework for mean-field corrections and fluctuations beyond bounded interactions.

Abstract

We study particle systems with singular pairwise interactions and non-vanishing diffusion in the mean-field scaling. A classical approach to describing corrections to mean-field behavior is through the analysis of correlation functions. For bounded interactions, the optimal estimates on correlations are well known: the $m$-particle correlation function is $G_{N,m}=O(N^{1-m})$ for all $m$. Such estimates, however, have remained out of reach for more singular interactions. In this work, we develop a new framework based on linearized correlation functions, which allows us to derive robust bounds for systems with merely square-integrable interaction kernels, providing the first systematic control of correlations in the singular setting. Although at first not optimal, our estimates can be partially refined a posteriori using the BBGKY hierarchy: in the case of bounded interactions, our method recovers the known optimal estimates with a simplified argument. As key applications, we establish the validity of the Bogolyubov correction to mean field and prove a central limit theorem for the empirical measure, extending these results beyond the bounded interaction regime for the first time.

Correlation estimates for Brownian particles with singular interactions

TL;DR

The paper develops a linearized correlation framework for second-order Langevin particle systems with singular, potentially unbounded interactions in the mean-field regime. By introducing linear correlations H_{N,m} and a tractable BBGKY-type hierarchy, it obtains robust L^2-type bounds and weak convergence results for G_{N,m} under square-integrable kernels, and extends the analysis to the overdamped case. The authors establish a Bogolyubov correction with propagation of chaos at rate N^{−1}, characterize a Gaussian CLT for empirical measure fluctuations via a linearized Dean–Kawasaki SPDE, and, in the bounded-forces regime, derive globally valid, sharper estimates with optimal scaling. These results significantly extend correlation control to singular interaction regimes and provide a rigorous framework for mean-field corrections and fluctuations beyond bounded interactions.

Abstract

We study particle systems with singular pairwise interactions and non-vanishing diffusion in the mean-field scaling. A classical approach to describing corrections to mean-field behavior is through the analysis of correlation functions. For bounded interactions, the optimal estimates on correlations are well known: the -particle correlation function is for all . Such estimates, however, have remained out of reach for more singular interactions. In this work, we develop a new framework based on linearized correlation functions, which allows us to derive robust bounds for systems with merely square-integrable interaction kernels, providing the first systematic control of correlations in the singular setting. Although at first not optimal, our estimates can be partially refined a posteriori using the BBGKY hierarchy: in the case of bounded interactions, our method recovers the known optimal estimates with a simplified argument. As key applications, we establish the validity of the Bogolyubov correction to mean field and prove a central limit theorem for the empirical measure, extending these results beyond the bounded interaction regime for the first time.

Paper Structure

This paper contains 14 sections, 11 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. General overview
  3. Main results
  4. Improved estimates for $K\in L^\infty$
  5. Tools for global hierarchical estimates
  6. New framework for correlation estimates
  7. Linear correlations
  8. BBGKY hierarchies
  9. Estimates on linear correlations
  10. Nonlinear correlations
  11. Weak limit of subcritically rescaled correlations
  12. Bogolyubov correction
  13. Fluctuations of empirical measure
  14. Case of bounded forces

Key Result

Theorem 1.1

Assume that $K$ belongs to $L^2((\mathbb T^d)^2)^d$ and derives from a potential, and further assume that the latter satisfies, for some $\beta>0$, where $W_-$ stands for the negative part of $W$. Assume that there is some $T_0>0$ and a weak solution $f$ of eq:VFP on $[0,T_0]$ with, for some $\beta>0$, Then there exist a time $T_*\in(0,T_0]$ and some $\beta_*>0$ such that for all $2\le m\le N$

Theorems & Definitions (23)

  • Theorem 1.1: Correlation estimates
  • Remark 1.2: Overdamped dynamics
  • Corollary 1.3: Bogolyubov correction
  • Corollary 1.4: CLT
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1: Global hierarchical estimates
  • proof
  • Remark 3.1
  • Lemma 3.2: Hierarchy for linear correlations
  • ...and 13 more