Ergodic Problems for Second-Order Mean Field Games with State Constraints

TL;DR

This paper investigates ergodic mean field games with state constraints, analyzing how agents' feedback controls influence the equilibrium, resulting in a second-order MFG system with boundary singularities and smooth densities.

Contribution

It characterizes the equilibrium in constrained mean field games with singular feedback controls, including the boundary behavior of the value function and density.

Findings

The value function blows up at the boundary.

The density of players remains smooth and flattens near the boundary.

The equilibrium solution may be unique.

Abstract

We study an ergodic mean field game problem with state constraints. In our model the agents are affected by idiosyncratic noise and use a (singular) feedback control to prevent the Brownian motion from exiting the domain. We characterize the equilibrium as the (possibly unique) solution to a second-order MFG system, where the value function blows up at the boundary while the density of the players is smooth and flattens near the boundary as a consequence of the singularity of the drift induced by the feedback strategy of the agents.