Hilbert Space-Valued LQG Mean Field Games: An Infinite-Dimensional Analysis

This paper presents a comprehensive study of Hilbert space-valued Linear-Quadratic-Gaussian (LQG) mean field games (MFGs), generalizing the classic LQG mean field game theory to scenarios where the state equations are driven by infinite-dimensional stochastic equations. In this framework, state and control processes take values in separable Hilbert spaces. Moreover, the state equations involve infinite dimensional noises, namely Q-Wiener processes. All agents are coupled through the average state of the population appearing in their linear dynamics and quadratic cost functional. In addition, the diffusion coefficient of each agent involves the state, control, and the average state processes. We first study the well-posedness of a system of coupled infinite-dimensional stochastic evolution equations, which forms the foundation of MFGs in Hilbert spaces. Next, we develop the Nash Certainty Equivalence principle and obtain a unique Nash equilibrium for the limiting Hilbert space-valued MFG. Finally, we establish the ε-Nash property for the finite-player game in Hilbert space.