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

This paper presents a comprehensive study of linear-quadratic (LQ) mean field games (MFGs) in Hilbert spaces, generalizing the classic LQ MFG theory to scenarios involving $N$ agents with dynamics governed by infinite-dimensional stochastic equations. In this framework, both state and control processes of each agent take values in separable Hilbert spaces. All agents are coupled through the average state of the population which appears in their linear dynamics and quadratic cost functional. Specifically, the dynamics of each agent incorporates an infinite-dimensional noise, namely a $Q$-Wiener process, and an unbounded operator. The diffusion coefficient of each agent is stochastic involving the state, control, and average state processes. We first study the well-posedness of a system of $N$ coupled semilinear infinite-dimensional stochastic evolution equations establishing the foundation of MFGs in Hilbert spaces. We then specialize to $N$-player LQ games described above and study the asymptotic behaviour as the number of agents, $N$, approaches infinity. We develop an infinite-dimensional variant of the Nash Certainty Equivalence principle and characterize a unique Nash equilibrium for the limiting MFG. Finally, we study the connections between the $N$-player game and the limiting MFG, demonstrating that the empirical average state converges to the mean field and that the resulting limiting best-response strategies form an $\epsilon$-Nash equilibrium for the $N$-player game in Hilbert spaces.