Ergodic Mean Field Games of Controls with State Constraints

TL;DR

This paper studies ergodic mean field games of controls with state constraints, establishing well-posedness of the associated second-order systems under certain conditions.

Contribution

It introduces a framework for analyzing ergodic MFGs with state constraints, proving well-posedness for systems with monotone coupling and quadratic Hamiltonians.

Findings

Systems are well-posed under monotone coupling.

Value functions blow up at the boundary, density vanishes at a matching rate.

Existence of solutions characterized by fixed-point relations.

Abstract

In a mean field game of controls, players seek to minimize a cost that depends on the joint distribution of players' states and controls. We consider an ergodic problem for second-order mean field games of controls with state constraints, in which equilibria are characterized by solutions to a second-order MFGC system where the value function blows up at the boundary, the density of players vanishes at a commensurate rate, and the joint distribution of states and controls satisfies the appropriate fixed-point relation. We prove that such systems are well-posed in the case of monotone coupling and Hamiltonians with at most quadratic growth.