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Approximation and perturbations of stable solutions to a stationary mean field game system

Jules Berry, Olivier Ley, Francisco J Silva

Abstract

This work introduces a new general approach for the numerical analysis of stable equilibria to second order mean field games systems in cases where the uniqueness of solutions may fail. For the sake of simplicity, we focus on a simple stationary case. We propose an abstract framework to study these solutions by reformulating the mean field game system as an abstract equation in a Banach space. In this context, stable equilibria turn out to be regular solutions to this equation, meaning that the linearized system is well-posed. We provide three applications of this property: we study the sensitivity analysis of stable solutions, establish error estimates for their finite element approximations, and prove the local converge of Newton's method in infinite dimensions.

Approximation and perturbations of stable solutions to a stationary mean field game system

Abstract

This work introduces a new general approach for the numerical analysis of stable equilibria to second order mean field games systems in cases where the uniqueness of solutions may fail. For the sake of simplicity, we focus on a simple stationary case. We propose an abstract framework to study these solutions by reformulating the mean field game system as an abstract equation in a Banach space. In this context, stable equilibria turn out to be regular solutions to this equation, meaning that the linearized system is well-posed. We provide three applications of this property: we study the sensitivity analysis of stable solutions, establish error estimates for their finite element approximations, and prove the local converge of Newton's method in infinite dimensions.
Paper Structure (16 sections, 22 theorems, 154 equations)

This paper contains 16 sections, 22 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. The mean field game system
  4. Well-posedness
  5. Stable solutions
  6. Reformulation of the MFG system
  7. Differentiability of the mapping $F$
  8. Differentiability in Hölder spaces
  9. Differentiability in Sobolev spaces
  10. Applications
  11. Stability under perturbations of the MFG system
  12. Finite Element approximation of the MFG system
  13. Newton's method
  14. Proof of \ref{['thm:well_posed']}
  15. Proof of \ref{['prop:isolated']}
  16. ...and 1 more sections

Key Result

Theorem 2.1

Assume that $f \in W^{1,\infty}(\mathbb{R})$. Then there exists a classical solution $(u,m) \in C^{2,\alpha}(\mathbb{T}^d) \times C^{2,\alpha}(\mathbb{T}^d)$ to eq:mfg, where the constant $\alpha$ is fixed in h:m0. Furthermore, if $f' \geq 0$ or if $\lambda$ is large enough, then this solution is un

Theorems & Definitions (43)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5: Stable solutions
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 33 more