Uniform-in-time propagation of chaos for the Cucker--Smale model

TL;DR

This work proves quantitative, uniform-in-time propagation of chaos for the Cucker–Smale model in the strongly interacting regime. The authors combine finite-time propagation-of-chaos results with a uniform-in-time stability estimate and a particle-duplication argument to derive explicit convergence rates in the number of particles for both empirical chaos and infinite-dimensional Wasserstein chaos. They obtain dimension-dependent rates for empirical chaos (e.g., $J^{-1/4}$, up to $J^{-1/d}$) and a Monte-Carlo–type rate of $J^{-1/2}$ for the full $J$-particle system in Wasserstein distance, all uniformly in time. The approach relies on a stability–consistency paradigm akin to numerical analysis, and is complemented by a synchronous-coupling construction and a detailed numerical illustration that confirms the uniform-boundedness of chaos indicators over time. These results advance the understanding of long-time behavior in flocking models and offer explicit, implementable rates for high-dimensional mean-field approximations.

Abstract

This paper presents an elementary proof of quantitative uniform-in-time propagation of chaos for the Cucker--Smale model under sufficiently strong interaction. The idea is to combine existing finite-time propagation of chaos estimates with existing uniform-in-time stability estimates for the interacting particle system, in order to obtain a uniform-in-time propagation of chaos estimate with an explicit rate of convergence in the number of particles. This is achieved via a method that is similar in spirit to the classical 'stability + consistency implies convergence' approach in numerical analysis.

Figures (3)

• Figure 1: Illustration of the decomposition \ref{['eq:decomp_global_error']}, which is the main idea of the proof of \ref{['theorem:uit_prop_of_chaos']}. Here, we used the notation $\mathcal{Z}^{J, n}_t = \Phi_{t - t_{n}} \circ \overline{\Phi}_{0, t_{n}} \bigl(\mathcal{Z}^{J}_0\bigr)$, for $t \geqslant t_n$, as well as $\overline {\mathcal{Z}}^J_t = \overline{\Phi}_{0, t} \bigl(\mathcal{Z}^{J}_0\bigr)$. The key idea of the proof is to break down the total error as a sum of small contributions $E_1, E_2, \dotsc$, each arising from the mismatch between the interacting particle system and the mean field particles over a bounded time interval $[t_i, t_{i+1}]$.
• Figure 2: Evolution of $\mathbf{E} \texttt{errX(t)}$ and $\mathbf{E} \texttt{errV(t)}$, as defined in \ref{['eq:errx']} and \ref{['eq:errv']}, for various values of $J$. We observe that $\mathbf{E} \texttt{errX(t)}$ initially increases, but tends to a constant in the limit $t \to \infty$ as flocking occurs.
• Figure 3: Value of $\mathbf{E} \texttt{errX(T)}$ for $T = 10$ and various values of the system size $J$.

Theorems & Definitions (19)

• Theorem 1.2
• Proposition 2.1: Uniform-in-time empirical chaos, with a rate
• Theorem 2.2: Uniform-in-time propagation of chaos, with a rate
• Remark 2.3: Comparison of this work with existing literature
• Remark 2.4: Alternative Cucker--Smale system
• Theorem 3.1: Finite-time propagation of chaos
• Theorem 3.2: Uniform-in-time stability estimate
• Lemma 3.3: Exponential concentration in velocity
• Corollary 4.1: Wasserstein stability for the empirical measures
• proof