Global Well-Posedness of Master Equations in Mean Field Games with Common Noise via Viscosity Solutions

This monograph establishes the global well-posedness of the Master Equation associated with Mean Field Games (MFG) subject to common noise. The Master Equation, a nonlinear partial differential equation defined on the Wasserstein space of probability measures, encapsulates the limit behavior of N-player differential games as N . We address the significant analytical challenges posed by the non-local nature of the coupling and the infinite-dimensional domain. By introducing a rigorous notion of viscosity solutions adapted to the geometry of the Wasserstein space P2(Rd), we overcome the lack of classical regularity typically inherent in second-order Hamilton-Jacobi-Bellman equations on measure spaces. We prove a comparison principle under structural monotonicity assumptions on the Hamiltonian and the coupling cost, which implies the uniqueness of the solution. Furthermore, we demonstrate the existence of solutions via the convergence of a vanishing viscosity approximation scheme or through the limit of the associated N-player Nash system. The results provide a robust probabilistic interpretation of the Master Equation in the presence of systemic noise.