LQ Mean Field Games with Common Noise in Hilbert Spaces: Small and Arbitrary Finite Time Horizons
Hanchao Liu, Dena Firoozi
TL;DR
We study linear-quadratic mean field games in Hilbert spaces with common noise driven by an infinite-dimensional Wiener process. The authors formulate the mean-field consistency as coupled forward-backward stochastic evolution equations and establish existence and uniqueness for mild solutions on small and arbitrary finite horizons using a decoupling field approach. They derive a Nash equilibrium for the limiting game via a Riccati equation and backward process, and prove an epsilon-Nash property for finite N with rates of order $N^{-1/2}$. The results extend previous Hilbert-space MFGs without common noise to include common noise and random diffusion, providing a foundation for infinite-dimensional MFGs with stochastic coefficients and broad applicability to SPDE-type models.
Abstract
We extend the results of (Liu and Firoozi, 2025), which develops the theory of linear-quadratic (LQ) mean field games in Hilbert spaces, by incorporating a common noise. This common noise is an infinite-dimensional Wiener process affecting the dynamics of all agents. In the presence of common noise, the mean-field consistency condition is characterized by a system of coupled forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces, whereas in its absence, it is represented by forward-backward deterministic evolution equations. We establish the existence and uniqueness of solutions to the coupled linear FBSEEs associated with the LQ MFG setting for small time horizons and prove the $ε$-Nash property of the resulting equilibrium strategy. Furthermore, for the first time in the literature, we develop an analysis that establishes the well-posedness of these coupled linear FBSEEs in Hilbert spaces, for which only mild solutions exist, over arbitrary finite time horizons.