Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The master equation for mean field game systems with fractional and nonlocal diffusions

Espen Robstad Jakobsen, Artur Rutkowski

TL;DR

This work addresses the master equation for mean field games driven by nonlocal Lévy diffusions on the whole space, proving existence and uniqueness of classical solutions under broad, physically relevant conditions. It develops a rigorous framework for derivatives with respect to probability measures, including the measure derivative $\frac{\delta U}{\delta m}$ and its spatial derivatives, and constructs a linearized forward–backward system to obtain regularity and stability results. The results extend prior local-diffusion theory to fractional and mixed local-nonlocal operators, allow the Hamiltonian to depend on the solution itself, and remove moment conditions on initial data, enabling a robust master-field formulation and a pathway toward N-player convergence. The paper also provides auxiliary regularity results for viscous Hamilton–Jacobi equations, parabolic equations, and FP systems in novel Hölder-dual settings, with careful treatment of measure-representable data and $L^1$-based compactness. Overall, it delivers a comprehensive, rigorous framework for master equations in nonlocal MFGs and lays groundwork for future convergence analyses and applications in systems with jump-diffusion noise.

Abstract

We prove existence and uniqueness of classical solutions of the master equation for mean field game (MFG) systems with fractional and nonlocal diffusions. We cover a large class of Lévy diffusions of order greater than one, including purely nonlocal, local, and even mixed local-nonlocal operators. In the process we prove refined well-posedness results for the MFG systems, results that include the mixed local-nonlocal case. We also show various auxiliary results on viscous Hamilton-Jacobi equations, linear parabolic equations, and linear forward-backward systems that may be of independent interest. This includes a rigorous treatment of certain equations and systems with data and solutions in the duals of Hölder spaces $C^γ_b$ on the whole of $\mathbb{R}^d$. We do not assume existence of any moments for the initial distributions of players. In a future work we will use the results of this paper to prove the convergence of $N$-player games to mean field games as $N\to\infty$.

The master equation for mean field game systems with fractional and nonlocal diffusions

TL;DR

This work addresses the master equation for mean field games driven by nonlocal Lévy diffusions on the whole space, proving existence and uniqueness of classical solutions under broad, physically relevant conditions. It develops a rigorous framework for derivatives with respect to probability measures, including the measure derivative and its spatial derivatives, and constructs a linearized forward–backward system to obtain regularity and stability results. The results extend prior local-diffusion theory to fractional and mixed local-nonlocal operators, allow the Hamiltonian to depend on the solution itself, and remove moment conditions on initial data, enabling a robust master-field formulation and a pathway toward N-player convergence. The paper also provides auxiliary regularity results for viscous Hamilton–Jacobi equations, parabolic equations, and FP systems in novel Hölder-dual settings, with careful treatment of measure-representable data and -based compactness. Overall, it delivers a comprehensive, rigorous framework for master equations in nonlocal MFGs and lays groundwork for future convergence analyses and applications in systems with jump-diffusion noise.

Abstract

We prove existence and uniqueness of classical solutions of the master equation for mean field game (MFG) systems with fractional and nonlocal diffusions. We cover a large class of Lévy diffusions of order greater than one, including purely nonlocal, local, and even mixed local-nonlocal operators. In the process we prove refined well-posedness results for the MFG systems, results that include the mixed local-nonlocal case. We also show various auxiliary results on viscous Hamilton-Jacobi equations, linear parabolic equations, and linear forward-backward systems that may be of independent interest. This includes a rigorous treatment of certain equations and systems with data and solutions in the duals of Hölder spaces on the whole of . We do not assume existence of any moments for the initial distributions of players. In a future work we will use the results of this paper to prove the convergence of -player games to mean field games as .
Paper Structure (36 sections, 37 theorems, 311 equations)

This paper contains 36 sections, 37 theorems, 311 equations.

Table of Contents

  1. Introduction
  2. Summary of the main results and contributions.
  3. Preliminaries
  4. Hölder and related function spaces
  5. Space of probability measures
  6. Derivative in the space of probability measures
  7. Operator $\mathcal{L}$ and its heat kernel
  8. Assumptions
  9. The Hamiltonian
  10. The operator
  11. Functions $F$ and $G$
  12. Monotonicity conditions
  13. Examples
  14. The operator $\mathcal{L}$
  15. The coupling terms $F$ and $G$
  16. ...and 21 more sections

Key Result

Lemma 2.3

Let $\rho\in C^{-\gamma}_b(\mathbb{R}^d)$ for some $\gamma \geq 0$ and assume that $f\in C^{\infty}(\mathbb{R}^d)$ with $f,Df,D^2f,\ldots\in L^1(\mathbb{R}^d)$. Then $\rho\ast f\in C^{-0}_b(\mathbb{R}^d)$. Furthermore, for $f=\eta_\varepsilon$ we have $\rho\ast\eta_\varepsilon \mathop{\longrightarro

Theorems & Definitions (96)

  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 86 more