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First-Order Mean Field Games with Time-Measurable and $C ^{1,1}$ Hamiltonians: Analysis and Approximation

Elisabetta Carlini, Valentina Coscetti, Francisco J. Silva · 2026 · arXiv

DOI: 10.48550/arXiv.2610.28335 · open-access PDF

Spanish

Introducimos y analizamos un marco numérico para una clase de sistemas de juego medio de campo no local de primer orden (MFG) con dos fuentes de baja regularidad: los datos dependientes del tiempo son meramente mensurables, y el Hamiltoniano es sólo de clase $C^{1,1}$ con respecto a la variable de impulso. El método propuesto combina una aproximación semi-Lagrangiana de la ecuación de Hamilton-Jacobi-Bellman con una discretización de Lagrange-Galerkin de la ecuación de continuidad.

Machine translation (Meta M2M100); the English original follows.

Abstract

We introduce and analyze a numerical framework for a class of first-order, non-local Mean Field Game (MFG) systems with two sources of low regularity: the time-dependent data are merely measurable, and the Hamiltonian is only of class $C^{1,1}$ with respect to the momentum variable. The proposed method combines a semi-Lagrangian approximation of the Hamilton--Jacobi--Bellman equation with a Lagrange--Galerkin discretization of the continuity equation. Temporal measurability is handled by averaging the data over the time-grid intervals and using the stability theory of $L^1$-viscosity solutions

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