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The Master Equation in a Bounded Domain with Absorption

Luca Di Persio, Matteo Garbelli, Michele Ricciardi

TL;DR

The paper establishes the well-posedness of the first-order Master Equation for Mean Field Games in a bounded domain with absorbing (Dirichlet) boundary conditions by formulating the problem on subprobability measures and analyzing the associated MFG system and a Dirichlet Fokker–Planck equation. A careful construction using negative Hölder spaces, a Schauder fixed-point framework, and Lasry–Lions monotonicity yields existence, uniqueness, and a priori bounds for the Master solution $U$, together with Lipschitz regularity of $U$ in the measure argument and a rigorous identification of the measure derivative $\delta U/\delta m=K$, where $K$ acts as a fundamental solution to the linearized system. The work also proves boundary-compatible Dirichlet conditions for $U$ and $\delta U/\delta m$, and derives strong regularity estimates essential for the Master Equation, including a detailed treatment of the Fokker–Planck equation in this setting. These results provide a foundational step toward convergence results for $N$-player games with absorption, to be pursued in future work.

Abstract

We analyze the Master Equation within Mean Field Games (MFG) theory considering a bounded domain with homogeneous Dirichlet conditions. Concerning the N-players differential game, the player's dynamic ends when touching the boundary. We analyze the well-posedness of the Master Equation and the regularity of its solutions for a suitable class of parabolic equations.

The Master Equation in a Bounded Domain with Absorption

TL;DR

The paper establishes the well-posedness of the first-order Master Equation for Mean Field Games in a bounded domain with absorbing (Dirichlet) boundary conditions by formulating the problem on subprobability measures and analyzing the associated MFG system and a Dirichlet Fokker–Planck equation. A careful construction using negative Hölder spaces, a Schauder fixed-point framework, and Lasry–Lions monotonicity yields existence, uniqueness, and a priori bounds for the Master solution , together with Lipschitz regularity of in the measure argument and a rigorous identification of the measure derivative , where acts as a fundamental solution to the linearized system. The work also proves boundary-compatible Dirichlet conditions for and , and derives strong regularity estimates essential for the Master Equation, including a detailed treatment of the Fokker–Planck equation in this setting. These results provide a foundational step toward convergence results for -player games with absorption, to be pursued in future work.

Abstract

We analyze the Master Equation within Mean Field Games (MFG) theory considering a bounded domain with homogeneous Dirichlet conditions. Concerning the N-players differential game, the player's dynamic ends when touching the boundary. We analyze the well-posedness of the Master Equation and the regularity of its solutions for a suitable class of parabolic equations.
Paper Structure (10 sections, 11 theorems, 169 equations)

This paper contains 10 sections, 11 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Notation and assumptions
  3. Hölder spaces
  4. Subprobability Measures and Generalized Wasserstein Measure
  5. Assumptions
  6. Well-posedness of the Master Equation
  7. The Fokker-Planck equation and the MFG system
  8. The Fokker-Planck equation
  9. Mean Field Games system and Lipschitz regularity of U
  10. Linearized system and differentiability of U with respect to the measure

Key Result

Lemma 2.1

The space $\mathcal{C}^{-(n+\alpha)}$ is a norm-closed subset of $\left(\mathcal{C}^{n+\alpha}\right)'$. Moreover, if $\rho\in\mathcal{C}^{-n}$, then there exist a sequence $\{\rho_k\}_k\subset\mathcal{C}^{2+\alpha,D}$ such that $\rho_k\to\rho$ in $\mathcal{C}^{-(n+\alpha)}$.

Theorems & Definitions (29)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • ...and 19 more