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Quantitative convergence for mean field control with common noise and degenerate idiosyncratic noise

Alekos Cecchin, Samuel Daudin, Joe Jackson, Mattia Martini

TL;DR

This work advances the quantitative understanding of mean-field control with common noise and degenerate idiosyncratic noise by deriving explicit convergence rates for the finite-particle value functions $V^N$ to the mean-field value $U$. The authors develop a robust regularization framework, including vanishing viscosity and mollification, and handle the challenging common-noise term via a transform that converts the problem into a finite-dimensional diffusion in an auxiliary variable, enabling precise subsolution estimates. The main contributions include rates that approach the conjectured $N^{-1/d}$ in favorable cases (e.g., constant diffusion or $U$ Lipschitz in $W^{-2, ext{∞}}$), and a general rate $N^{-1/(3d+19)}$ when degeneracy and non-constancy of idiosyncratic noise interact with common noise; the zero-noise analysis further highlights control-sensitivity phenomena and the role of convexity in achieving equality. Together, these results refine the understanding of how noise structure influences convergence and provide practical benchmarks for numerical approximations in high-dimensional, noisy mean-field control problems.

Abstract

We consider the convergence problem in the setting of mean field control with common noise and degenerate idiosyncratic noise. Our main results establish a rate of convergence of the finite-dimensional value functions $V^N$ towards the mean field value function $U$. In the case that the idiosyncratic noise is constant (but possibly degenerate), we obtain the rate $N^{-1/(d+7)}$, which is close to the conjectured optimal rate $N^{-1/d}$, and improves on the existing literature even in the non-degenerate setting. In the case that the idiosyncratic noise can be both non-constant and degenerate, the argument is more complicated, and we instead find the rate $N^{-1/(3d + 19)}$. Our proof strategy builds on the one initiated in [Daudin, Delarue, Jackson - JFA, 2024] in the case of non-degenerate idiosyncratic noise and zero common noise, which consists of approximating $U$ by more regular functions which are almost subsolutions of the infinite-dimensional Hamilton-Jacobi equation solved by $U$. Because of the different noise structure, several new steps are necessary in order to produce an appropriate mollification scheme. In addition to our main convergence results, we investigate the case of zero idiosyncratic noise, and show that sharper results can be obtained there by purely control-theoretic arguments. We also provide examples to demonstrate that the value function is sensitive to the choice of admissible controls in the zero noise setting.

Quantitative convergence for mean field control with common noise and degenerate idiosyncratic noise

TL;DR

This work advances the quantitative understanding of mean-field control with common noise and degenerate idiosyncratic noise by deriving explicit convergence rates for the finite-particle value functions to the mean-field value . The authors develop a robust regularization framework, including vanishing viscosity and mollification, and handle the challenging common-noise term via a transform that converts the problem into a finite-dimensional diffusion in an auxiliary variable, enabling precise subsolution estimates. The main contributions include rates that approach the conjectured in favorable cases (e.g., constant diffusion or Lipschitz in ), and a general rate when degeneracy and non-constancy of idiosyncratic noise interact with common noise; the zero-noise analysis further highlights control-sensitivity phenomena and the role of convexity in achieving equality. Together, these results refine the understanding of how noise structure influences convergence and provide practical benchmarks for numerical approximations in high-dimensional, noisy mean-field control problems.

Abstract

We consider the convergence problem in the setting of mean field control with common noise and degenerate idiosyncratic noise. Our main results establish a rate of convergence of the finite-dimensional value functions towards the mean field value function . In the case that the idiosyncratic noise is constant (but possibly degenerate), we obtain the rate , which is close to the conjectured optimal rate , and improves on the existing literature even in the non-degenerate setting. In the case that the idiosyncratic noise can be both non-constant and degenerate, the argument is more complicated, and we instead find the rate . Our proof strategy builds on the one initiated in [Daudin, Delarue, Jackson - JFA, 2024] in the case of non-degenerate idiosyncratic noise and zero common noise, which consists of approximating by more regular functions which are almost subsolutions of the infinite-dimensional Hamilton-Jacobi equation solved by . Because of the different noise structure, several new steps are necessary in order to produce an appropriate mollification scheme. In addition to our main convergence results, we investigate the case of zero idiosyncratic noise, and show that sharper results can be obtained there by purely control-theoretic arguments. We also provide examples to demonstrate that the value function is sensitive to the choice of admissible controls in the zero noise setting.
Paper Structure (30 sections, 44 theorems, 346 equations)

This paper contains 30 sections, 44 theorems, 346 equations.

Table of Contents

  1. Introduction
  2. The $N$-particle and mean field control problems
  3. Existing results on the convergence problem
  4. Our contribution
  5. Strategy of the proof
  6. The degenerate and non-constant idiosyncratic noise
  7. The common noise
  8. Summary of the regularization argument
  9. Possible extensions and limitations of the method
  10. Organization of the paper
  11. Notation, preliminaries, and main results
  12. Problem statement
  13. Assumptions and main results
  14. The vanishing viscosity procedure
  15. Definition of $V^{N,\eta}$
  16. ...and 15 more sections

Key Result

Lemma 2.1

Let $\Phi, \Psi : \mathcal{P}(\mathbb{T}^d) \to \mathbb{R}$, and assume that that $\Psi \in \mathcal{C}^1(\mathcal{P}(T^d))$. Assume moreover that for some $m_0 \in \mathcal{P}(\mathbb{T}^d)$ satisfying $m_0 \geq c\mathrm{Leb}$ for some $c > 0$, we have Then, As a consequence, if $\Psi \in \mathcal{C}^1(\mathcal{P}(T^d))$ is Lipschitz with respect to $W^{-2,\infty}$, then

Theorems & Definitions (87)

  • Lemma 2.1
  • Remark 2.3
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • ...and 77 more