The ergodic Mean Field Game system for a type of state constraint condition

TL;DR

This paper studies the well-posedness of a state-constrained ergodic Mean Field Game system with boundary conditions in bounded domains, establishing existence and uniqueness results for superlinear Hamiltonians.

Contribution

It introduces a novel analysis of ergodic Mean Field Game systems with infinite boundary conditions, covering both local and non-local couplings, and proves key existence and uniqueness results.

Findings

Existence of solutions under superlinear Hamiltonians.

Uniqueness of solutions using boundary decay properties.

Applicability to both local and non-local coupling scenarios.

Abstract

This paper investigates the well-posedness of a type of state constraint ergodic Mean Field Game system in a bounded domain in which the Hamilton-Jacobi-Bellman equation is paired with an infinite Dirichlet boundary condition. In this setting, the infinite boundary condition prevents the underlying stochastic trajectories from reaching the boundary, resulting in a state constraint solution. We consider a class of superlinear power-like Hamiltonians and establish existence and uniqueness results under appropriate assumptions on the coupling. We treat both non-local and local couplings, and establish the uniqueness of the system by exploiting the decay of the density near the boundary.