Linear quadratic Mean Field Games and Master Equations in Hilbert spaces with common noise

Abstract

We study linear-quadratic Mean Field Games with common noise in an infinite-dimensional Hilbert space. The state dynamics are driven by a possibly unbounded linear operator, and the interaction with the population enters through the mean of its conditional distribution, both in the dynamics and in the cost functional. We allow general quadratic costs including linear and non-separable terms. Exploiting the linear-quadratic structure, we characterize mild solutions of the Mean Field Game system and of the associated Master Equation through systems of operator-valued Riccati equations, linear ordinary differential equations, and stochastic differential equations. A central difficulty is the analysis of a possibly non-self-adjoint Riccati equation arising from the coupling with the population mean. Under suitable structural assumptions, we establish global-in-time existence, uniqueness, and uniform a priori estimates for the corresponding mild solutions. The analysis relies on a fixed-point argument, stability estimates, and Yosida approximations of the unbounded generator. We also prove that the solution of the Master Equation, evaluated along the equilibrium flow of conditional distributions, recovers the Mean Field Game value function, which is not obvious due to the mild formulation used for both solutions. Finally, a verification theorem shows that the mild Mean Field Game solution coincides with the value function of the representative agent and identifies the optimal feedback control.