Equivalence for Wasserstein-Space Monotone Solutions

Establish the equivalence between Wasserstein-space formulations of monotone solutions and weak solutions for the mean field game of controls master equation, analogous to the equivalence established in the Hilbert-space formulation.

Background

The paper establishes equivalence between its weak-solution notion and a Hilbert-space formulation of monotone solutions for the mean field game of controls master equation, under Lasry–Lions monotonicity and displacement λ-monotonicity. The authors note that existing monotone-solution formulations based on Wasserstein space differ from the Hilbert-space approach, particularly because of the treatment of common noise and the associated probability-measure spaces.

The unresolved problem is to prove analogous equivalence results when monotone solutions are formulated directly on Wasserstein space rather than through the Hilbert-space lifting method. Such a result would extend the paper’s unification of nonsmooth solution notions to the Wasserstein-space frameworks used in related mean field game literature.