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A particle system approach towards the global well-posedness of master equations for potential mean field games of control

Huafu Liao, Chenchen Mou

TL;DR

This work addresses the global well-posedness of HJB/master equations for generalized mean field control and potential mean field games of control by combining a particle-system analysis with probabilistic mean-field techniques. The authors construct local-in-time solutions via a mean-field stochastic maximum principle, obtain uniform a priori estimates through displacement convexity, and bootstrap them to global well-posedness, including degenerate-noise limits. They establish propagation of chaos with explicit convergence rates, relate the particle-system decoupling field to the master-field, and derive Lipschitz approximations to optimal feedbacks along with a quantitatively justified approximate Nash equilibrium for finite N. The results provide a rigorous bridge between finite-particle control problems and their mean-field limits, with direct implications for numerical schemes and stability analyses in generalized MFC and MFGC settings.

Abstract

This paper studies the $N$-particle systems as well as the HJB/master equations for a class of generalized mean field control (MFC) problems and the corresponding potential mean field games of control (MFGC). A local in time classical solution for the HJB equation is generated via a probabilistic approach based on the mean field maximum principle. Given an extension of the so called displacement convexity condition, we obtain the uniform estimates on the HJB equation for the $N$-particle system. Such estimates imply the displacement convexity/semi-concavity and thus the prior estimates on the solution to the HJB equation for generalized MFC problems. The global well-posedness of HJB/master equation for generalized MFC/potential MFGC is then proved thanks to the local well-posedness and the prior estimates. In view of the nature of the displacement convexity condition, such well-posedness is also true for the degenerated case. Our analysis on the $N$-particle system also induces an Lipschitz approximator to the optimal feedback function in generalized MFC/potential MFGC where an algebraic convergence rate is obtained. Furthermore, an alternative approximate Nash equilibrium is proposed based on the $N$-particle system, where the approximation error is quantified thanks to the aforementioned uniform estimates.

A particle system approach towards the global well-posedness of master equations for potential mean field games of control

TL;DR

This work addresses the global well-posedness of HJB/master equations for generalized mean field control and potential mean field games of control by combining a particle-system analysis with probabilistic mean-field techniques. The authors construct local-in-time solutions via a mean-field stochastic maximum principle, obtain uniform a priori estimates through displacement convexity, and bootstrap them to global well-posedness, including degenerate-noise limits. They establish propagation of chaos with explicit convergence rates, relate the particle-system decoupling field to the master-field, and derive Lipschitz approximations to optimal feedbacks along with a quantitatively justified approximate Nash equilibrium for finite N. The results provide a rigorous bridge between finite-particle control problems and their mean-field limits, with direct implications for numerical schemes and stability analyses in generalized MFC and MFGC settings.

Abstract

This paper studies the -particle systems as well as the HJB/master equations for a class of generalized mean field control (MFC) problems and the corresponding potential mean field games of control (MFGC). A local in time classical solution for the HJB equation is generated via a probabilistic approach based on the mean field maximum principle. Given an extension of the so called displacement convexity condition, we obtain the uniform estimates on the HJB equation for the -particle system. Such estimates imply the displacement convexity/semi-concavity and thus the prior estimates on the solution to the HJB equation for generalized MFC problems. The global well-posedness of HJB/master equation for generalized MFC/potential MFGC is then proved thanks to the local well-posedness and the prior estimates. In view of the nature of the displacement convexity condition, such well-posedness is also true for the degenerated case. Our analysis on the -particle system also induces an Lipschitz approximator to the optimal feedback function in generalized MFC/potential MFGC where an algebraic convergence rate is obtained. Furthermore, an alternative approximate Nash equilibrium is proposed based on the -particle system, where the approximation error is quantified thanks to the aforementioned uniform estimates.

Paper Structure

This paper contains 16 sections, 32 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Notations, assumptions and main results
  3. The propagation of chaos and the verification results
  4. The generation of a local in time solution
  5. A priori estimates on $V_N$ and $V$
  6. A priori estimates on the first order derivatives of $V_N$
  7. A priori estimates on the second order derivatives of $V_N$
  8. A priori estimates on the second order derivatives of $V$
  9. A priori estimates on higher order derivatives of $V$
  10. Well-posedness and Feynman-Kac representation of linear master equations
  11. A priori estimates on higher order derivatives
  12. The global well-posedness of HJB/master equations
  13. The global well-posedness of HJB equation \ref{['HJB-MF']}
  14. The global well-posedness of master equation \ref{['HJB-MFGC']}
  15. The degenerated case
  16. ...and 1 more sections

Key Result

Lemma 2.2

Assume Assumption assumption, then as well as

Theorems & Definitions (68)

  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Proposition 3.1
  • proof
  • ...and 58 more