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On uniform in time propagation of chaos in metastable cases: the Curie-Weiss model

Lucas Journel, Pierre Le Bris

TL;DR

Addressing metastability in mean-field dynamics, this paper analyzes uniform-in-time propagation of chaos for the Curie–Weiss model under a positive-magnetization conditioning. It combines a killed Markov process and a Doob $h$-transform to relate short-time PoC to long-time quasi-stationary behavior, proving existence and quantitative convergence to a QSD and deriving an explicit rate of convergence to the nonlinear limit within a metastable basin. The key contributions are a finite-time PoC bound, a long-time QSD convergence with exponential rate, and a uniform-in-time approximation of the conditioned empirical measure by the nonlinear limit up to a polynomial time scale $n^{\alpha}$, plus a mechanism to bridge times $t$ between small and large regimes. The results illustrate that, within a metastable basin, the conditioned particle system can be accurately described by the nonlinear mean-field limit, with explicit $n$-dependent rates, suggesting applicability to other metastable mean-field models and guiding future study of SDE/McKean–Vlasov systems at low noise.

Abstract

Many low temperature particle systems in mean-field interaction are ergodic with respect to a unique invariant measure, while their (non-linear) mean-field limit may possess several steady states. In particular, in such cases, propagation of chaos (i.e. the convergence of the particle system to its mean-field limit as n, the number of particles, goes to infinity) cannot hold uniformly in time since the long-time behaviors of the two processes are a priori incompatible. However, the particle system may be metastable, and the time needed to exit the basin of attraction of one of the steady states of its limit, and go to another, is exponentially (in n) long. Before this exit time, the particle system reaches a (quasi-)stationary distribution, which we expect to be a good approximation of the corresponding non-linear steady state. Our goal is to study the typical metastable behavior of the empirical measure of such mean-field systems, starting in this work with the Curie-Weiss model. We thus show uniform in time propagation of chaos of the spin system conditioned to keeping a positive magnetization.

On uniform in time propagation of chaos in metastable cases: the Curie-Weiss model

TL;DR

Addressing metastability in mean-field dynamics, this paper analyzes uniform-in-time propagation of chaos for the Curie–Weiss model under a positive-magnetization conditioning. It combines a killed Markov process and a Doob -transform to relate short-time PoC to long-time quasi-stationary behavior, proving existence and quantitative convergence to a QSD and deriving an explicit rate of convergence to the nonlinear limit within a metastable basin. The key contributions are a finite-time PoC bound, a long-time QSD convergence with exponential rate, and a uniform-in-time approximation of the conditioned empirical measure by the nonlinear limit up to a polynomial time scale , plus a mechanism to bridge times between small and large regimes. The results illustrate that, within a metastable basin, the conditioned particle system can be accurately described by the nonlinear mean-field limit, with explicit -dependent rates, suggesting applicability to other metastable mean-field models and guiding future study of SDE/McKean–Vlasov systems at low noise.

Abstract

Many low temperature particle systems in mean-field interaction are ergodic with respect to a unique invariant measure, while their (non-linear) mean-field limit may possess several steady states. In particular, in such cases, propagation of chaos (i.e. the convergence of the particle system to its mean-field limit as n, the number of particles, goes to infinity) cannot hold uniformly in time since the long-time behaviors of the two processes are a priori incompatible. However, the particle system may be metastable, and the time needed to exit the basin of attraction of one of the steady states of its limit, and go to another, is exponentially (in n) long. Before this exit time, the particle system reaches a (quasi-)stationary distribution, which we expect to be a good approximation of the corresponding non-linear steady state. Our goal is to study the typical metastable behavior of the empirical measure of such mean-field systems, starting in this work with the Curie-Weiss model. We thus show uniform in time propagation of chaos of the spin system conditioned to keeping a positive magnetization.

Paper Structure

This paper contains 9 sections, 3 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. About metastability and propagation of chaos
  3. The Curie-Weiss model and main results
  4. Outline
  5. Future directions.
  6. Organization of the paper.
  7. Notations.
  8. Some intermediate results
  9. Results on the Curie-Weiss model

Key Result

Proposition 1.1

Denote $\mathcal{L}(m_t^n)$ (resp. $\mathcal{L}(\overline{m}_t)$) the law of the process $m_t^n$ (resp. $\overline{m}_t$ for which, even though it is deterministic, we allow for a random initial condition $\overline{m}_0$). There exist $c,C>0$ such that for all $t\geqslant 0$ and $n\in\mathbb{N}$ Furthermore, if $\beta<1$, there exist $c,C>0$ such that for all $t\geqslant0$ and for all $n\in\mathb

Figures (1)

  • Figure 1: Left: Potential $g$ defined in \ref{['eq:pot-CW']} for $\beta=1.2$. There are two minimizers denoted $m_-$ and $m_+$. Right: The process is conditioned to staying in $[\varepsilon,1]$.

Theorems & Definitions (5)

  • Proposition 1.1: Finite time PoC
  • Theorem 1: Long-time convergence to QSD
  • Theorem 2
  • Remark 1.1
  • proof : Proof of Proposition \ref{['prop:PoCnonUnif']}