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

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