MONOTONE SOLUTIONS OF THE MASTER EQUATION FOR MEAN FIELD GAMES WITH IDIOSYNCRATIC NOISE

We introduce a notion of a weak solution of the master equation without idiosyncratic noise in mean field game theory and establish its existence, uniqueness up to a constant, and consistency with classical solutions when it is smooth. We work in a monotone setting and rely on Lions' Hilbert space approach. For the first-order master equation without idiosyncratic noise, we also give an equivalent definition in the space of measures and establish the well-posedness.