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Uniform in time weak propagation of chaos on the torus

François Delarue, Alvin Tse

TL;DR

The paper tackles the problem of uniform-in-time weak propagation of chaos for weakly interacting diffusions on the torus, establishing an $O(1/N)$ bound between the empirical measure and the McKean–Vlasov law for all times. It develops a master-equation approach, relies on the linearization of the forward equation, and derives ergodic/tangent-process estimates to achieve time-uniform control under generic regularity assumptions, including small mean-field interactions and $H$-stable potentials. The authors extend the framework to models without a unique invariant measure, notably the super-critical Kuramoto model, where they obtain a uniform bound for rotation-invariant test functionals despite multiple equilibria and metastability. The work provides a flexible PDE-based strategy that couples master-equation analysis with ergodic theory to yield robust uniform-in-time propagation results with potential implications for mean-field models in bounded geometries and related dynamical systems.

Abstract

We address the long time behaviour of weakly interacting diffusive particle systems on the d-dimensional torus. Our main result is to show that, under certain regularity conditions, the weak error between the empirical distribution of the particle system and the theoretical law of the limiting process (governed by a McKean-Vlasov stochastic differential equation) is of the order O(1/N), uniform in time on [0, infinity), where N is the number of particles in the interacting diffusion. This comprises general interaction terms with a small enough mean-field dependence together with interactions terms driven by an H-stable potential. Our approach relies on a systematic analysis of the long-time behaviour of the derivatives of the semigroup generated by the McKean-Vlasov SDE, which may be explicitly computed through the linearised Fokker-Planck equation. Ergodic estimates for the latter hence play a key role in our analysis. We believe that this strategy is flexible enough to cover a wider broad of situations. To wit, we succeed in adapting it to the super-critical Kuramoto model, for which the corresponding McKean-Vlasov equation has several invariant measures.

Uniform in time weak propagation of chaos on the torus

TL;DR

The paper tackles the problem of uniform-in-time weak propagation of chaos for weakly interacting diffusions on the torus, establishing an bound between the empirical measure and the McKean–Vlasov law for all times. It develops a master-equation approach, relies on the linearization of the forward equation, and derives ergodic/tangent-process estimates to achieve time-uniform control under generic regularity assumptions, including small mean-field interactions and -stable potentials. The authors extend the framework to models without a unique invariant measure, notably the super-critical Kuramoto model, where they obtain a uniform bound for rotation-invariant test functionals despite multiple equilibria and metastability. The work provides a flexible PDE-based strategy that couples master-equation analysis with ergodic theory to yield robust uniform-in-time propagation results with potential implications for mean-field models in bounded geometries and related dynamical systems.

Abstract

We address the long time behaviour of weakly interacting diffusive particle systems on the d-dimensional torus. Our main result is to show that, under certain regularity conditions, the weak error between the empirical distribution of the particle system and the theoretical law of the limiting process (governed by a McKean-Vlasov stochastic differential equation) is of the order O(1/N), uniform in time on [0, infinity), where N is the number of particles in the interacting diffusion. This comprises general interaction terms with a small enough mean-field dependence together with interactions terms driven by an H-stable potential. Our approach relies on a systematic analysis of the long-time behaviour of the derivatives of the semigroup generated by the McKean-Vlasov SDE, which may be explicitly computed through the linearised Fokker-Planck equation. Ergodic estimates for the latter hence play a key role in our analysis. We believe that this strategy is flexible enough to cover a wider broad of situations. To wit, we succeed in adapting it to the super-critical Kuramoto model, for which the corresponding McKean-Vlasov equation has several invariant measures.

Paper Structure

This paper contains 32 sections, 40 theorems, 330 equations.

Table of Contents

  1. Introduction
  2. State of the art
  3. Our contribution.
  4. Organisation and notations
  5. Main method of proof in this paper
  6. Master equation
  7. Expansion along the particle system
  8. Main assumptions
  9. Functional spaces
  10. Main assumptions
  11. Examples
  12. Linear interaction
  13. A completely non-linear example
  14. Uniform weak propagation of chaos for McKean-Vlasov equations
  15. Second order mixed spatial derivatives of the second order linear functional derivative of U
  16. ...and 17 more sections

Key Result

Proposition 2.1

If $b$ and $\Phi$ are smooth enough then, for every $i,j \in \{1, \ldots, d\}$, the derivatives exist and are globally Lipschitz continuous w.r.t. $(\mu,y_{1},y_{2})$ for the Euclidean and ${\mathcal{W}}_1$ norms, uniformly in time $t$ in compact subsets, and are continuous in time (and hence jointly continuous).

Theorems & Definitions (79)

  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 69 more