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Higher-order propagation of chaos in $L^2$ for interacting diffusions

Elias Hess-Childs, Keefer Rowan

TL;DR

The paper establishes propagation of chaos for interacting diffusions in a strong $L^2$-based metric on arbitrary time horizons, using a BBGKY hierarchy and a cluster-expansion perturbation framework. It constructs higher-order corrections $f^i_j$ through a hierarchical perturbative scheme built from cluster functions $g^i_j$, and proves that $f_{j,N}$ approximates $\sum_{k=0}^i N^{-k} f^k_j$ with error $O(N^{-(i+1)})$ in the $L^2$-weighted sense, for $j$ up to roughly $N^{2/3}$. The analysis yields an optimal $N^{-1}$ rate at leading order, with $i\ge1$ providing computable, low-dimensional approximations to the $j$-particle marginals. The approach generalizes from a torus setting to the full space by a careful truncation, periodization, and limiting argument, ensuring the results hold in $\mathbb{R}^d$ as well. Overall, the work advances quantitative chaos propagation results by leveraging $L^2$-type controls and perturbative cluster methods to access higher-order corrections with rigorous error bounds.

Abstract

In this paper, we study diffusions with bounded pairwise interaction. We show for the first time propagation of chaos on arbitrary time horizons in a stronger $L^2$-based distance, as opposed to the usual Wasserstein or relative entropy distances. The estimate is based on iterating inequalities derived from the BBGKY hierarchy and does not follow directly from bounds on the full $N$-particle density. This argument gives the optimal rate in $N$, showing the distance between the $j$-particle marginal density and the tensor product of the mean-field limit is $O(N^{-1})$. We use cluster expansions to give perturbative higher-order corrections to the mean-field limit. For an arbitrary order $i$, these provide ``low-dimensional'' approximations to the $j$-particle marginal density with error $O(N^{-(i+1)})$.

Higher-order propagation of chaos in $L^2$ for interacting diffusions

TL;DR

The paper establishes propagation of chaos for interacting diffusions in a strong -based metric on arbitrary time horizons, using a BBGKY hierarchy and a cluster-expansion perturbation framework. It constructs higher-order corrections through a hierarchical perturbative scheme built from cluster functions , and proves that approximates with error in the -weighted sense, for up to roughly . The analysis yields an optimal rate at leading order, with providing computable, low-dimensional approximations to the -particle marginals. The approach generalizes from a torus setting to the full space by a careful truncation, periodization, and limiting argument, ensuring the results hold in as well. Overall, the work advances quantitative chaos propagation results by leveraging -type controls and perturbative cluster methods to access higher-order corrections with rigorous error bounds.

Abstract

In this paper, we study diffusions with bounded pairwise interaction. We show for the first time propagation of chaos on arbitrary time horizons in a stronger -based distance, as opposed to the usual Wasserstein or relative entropy distances. The estimate is based on iterating inequalities derived from the BBGKY hierarchy and does not follow directly from bounds on the full -particle density. This argument gives the optimal rate in , showing the distance between the -particle marginal density and the tensor product of the mean-field limit is . We use cluster expansions to give perturbative higher-order corrections to the mean-field limit. For an arbitrary order , these provide ``low-dimensional'' approximations to the -particle marginal density with error .
Paper Structure (23 sections, 26 theorems, 364 equations, 2 figures)

This paper contains 23 sections, 26 theorems, 364 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Overview of the argument
  4. Higher-order corrections to propagation of chaos
  5. $L^2$ hierarchy estimates
  6. Organization of the argument
  7. Acknowledgments
  8. Cluster expansions and perturbation theory
  9. Cluster expansion
  10. Perturbative expansion of the cluster functions
  11. Existence of the $g^i_j$
  12. Hierarchy bounds
  13. ODE hierarchy estimates
  14. Proof of a special case of Theorem \ref{['thm.main-result-intro']}
  15. Proofs of $g^i_j$ marginalization
  16. ...and 8 more sections

Key Result

Proposition 1.2

Suppose the initial distribution $f \in L^1(\Omega)$ and the interaction $K \in L_\delta^\infty(\Omega^{2})$. Then there exists a family of functions $g^i_j \in C^0_{loc}([0,\infty), L^1(\Omega^{j})),$ where $j \in \{1,2,\dotsc\}$ and $i \in \{0,1,\dotsc\}$ so that $g^i_j$ solve the equations eq.g-i

Figures (2)

  • Figure 1.1: The dependency graph for $g_j^i$.
  • Figure 1.2: The dependency graph for $f_j^i$.

Theorems & Definitions (79)

  • Definition
  • Definition
  • Definition
  • Definition
  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • ...and 69 more