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Quantitative Propagation of Chaos for Singular Interacting Particle Systems Driven by Fractional Brownian Motion

Lucio Galeati, Khoa Lê, Avi Mayorcas

TL;DR

This work analyzes N-particle systems driven by fractional Brownian motion with highly singular, possibly distributional interactions. It develops a robust framework combining Sznitman coupling and stochastic sewing to obtain quantitative propagation of chaos and mean-field convergence to McKean–Vlasov equations, achieving the sharp $N^{-1/2}$ rate under Besov-type regularity controlled by the Hurst parameter $H$ and a quasi-Lipschitz index $ obreak obreak obreak abla^ obreak abla$ constraint. The authors establish well-posedness for both the IPS and the MKV equations in regimes of distributional and functional drifts by approximating with smooth drifts and passing to the limit, thereby extending stochastic regularization by noise to singular interactions. They further discuss potential directions for fluctuations, large deviations, multiplicative noise, and long-time behavior, highlighting the broad relevance of their regularization-by-noise approach for non-Markovian stochastic systems in physics and applied mathematics.

Abstract

We consider interacting systems particle driven by i.i.d. fractional Brownian motions, subject to irregular, possibly distributional, pairwise interactions. We show propagation of chaos and mean field convergence to the law of the associated McKean--Vlasov equation, as the number of particles $N\to\infty$, with quantitative sharp rates of order $N^{-1/2}$. Our results hold for a wide class of possibly time-dependent interactions, which are only assumed to satisfy a Besov-type regularity, related to the Hurst parameter $H\in (0,+\infty)\setminus \mathbb{N}$ of the driving noises. In particular, as $H$ decreases to $0$, interaction kernels of arbitrary singularity can be considered, a phenomenon frequently observed in regularization by noise results. Our proofs rely on a combinations of Sznitman's direct comparison argument with stochastic sewing techniques.

Quantitative Propagation of Chaos for Singular Interacting Particle Systems Driven by Fractional Brownian Motion

TL;DR

This work analyzes N-particle systems driven by fractional Brownian motion with highly singular, possibly distributional interactions. It develops a robust framework combining Sznitman coupling and stochastic sewing to obtain quantitative propagation of chaos and mean-field convergence to McKean–Vlasov equations, achieving the sharp rate under Besov-type regularity controlled by the Hurst parameter and a quasi-Lipschitz index constraint. The authors establish well-posedness for both the IPS and the MKV equations in regimes of distributional and functional drifts by approximating with smooth drifts and passing to the limit, thereby extending stochastic regularization by noise to singular interactions. They further discuss potential directions for fluctuations, large deviations, multiplicative noise, and long-time behavior, highlighting the broad relevance of their regularization-by-noise approach for non-Markovian stochastic systems in physics and applied mathematics.

Abstract

We consider interacting systems particle driven by i.i.d. fractional Brownian motions, subject to irregular, possibly distributional, pairwise interactions. We show propagation of chaos and mean field convergence to the law of the associated McKean--Vlasov equation, as the number of particles , with quantitative sharp rates of order . Our results hold for a wide class of possibly time-dependent interactions, which are only assumed to satisfy a Besov-type regularity, related to the Hurst parameter of the driving noises. In particular, as decreases to , interaction kernels of arbitrary singularity can be considered, a phenomenon frequently observed in regularization by noise results. Our proofs rely on a combinations of Sznitman's direct comparison argument with stochastic sewing techniques.
Paper Structure (22 sections, 33 theorems, 265 equations)

This paper contains 22 sections, 33 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Main Ideas of the Proof
  3. Context and Wider Literature
  4. Further Directions and Open Problems
  5. Structure of the Paper
  6. Notation, Conventions and Setup
  7. Preliminaries
  8. Fractional Brownian motion, local nondeterminism and conditional norms
  9. Controls and spaces of bounded variation
  10. Stochastic sewing
  11. Integral estimates
  12. The equations with regular drifts
  13. A Priori and Stability Estimates for Interacting Particle Systems
  14. A Priori and Stability Estimates for McKean--Vlasov Equations
  15. Propagation of chaos estimates
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $T\in (0,\infty)$, $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space carrying $\{X^{i}_0\}_{i=1}^\infty, X_0$, a collection of i.i.d random variables in $\mathbb{R}^d$, and $\{W^{i,H}\}_{i=1}^\infty, W^H$, a collection of i.i.d fractional Brownian motions of Hurst parameter $H\in (0,+\inf Then the following hold:

Theorems & Definitions (96)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Example 1.10: Convolutional, Pointwise and Statistical Interactions
  • ...and 86 more