Infinite-Dimensional LQ Mean Field Games with Common Noise: Small and Arbitrary Finite Time Horizons

TL;DR

This paper extends the theory of infinite-dimensional linear-quadratic mean field games to include common noise, establishing existence, uniqueness, and well-posedness of solutions over small and arbitrary finite time horizons, and proving the epsilon-Nash equilibrium property.

Contribution

It introduces the first well-posedness results for a class of infinite-dimensional linear forward-backward stochastic evolution equations over arbitrary finite horizons, with applications to mean field games with common noise.

Findings

Established existence and uniqueness of solutions for small time horizons.

Proved well-posedness of coupled FBSEEs for arbitrary finite horizons.

Demonstrated the epsilon-Nash property of equilibrium strategies.

Abstract

We extend the results of (Liu and Firoozi, 2025), which develops the theory of linear-quadratic (LQ) mean field games (MFGs) in Hilbert spaces, by incorporating a common noise. This common noise is modeled as 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 coupled forward-backward deterministic evolution equations. We establish the existence and uniqueness of solutions to the coupled linear FBSEEs associated with the LQ MFG framework for small time horizons and prove the $\epsilon$-Nash property of the resulting equilibrium strategy. Furthermore, we establish the well-posedness of these coupled linear FBSEEs for arbitrary finite time horizons. Beyond the specific context of MFGs, our analysis also yields a broader contribution by providing, to the best of our knowledge, the first well-posedness result for a class of infinite-dimensional linear FBSEEs, for which only mild solutions exist, over arbitrary finite time horizons.