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LQ Mean Field Games with Common Noise in Hilbert Spaces: Small and Arbitrary Finite Time Horizons

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  • Hanchao Liu
  • Dena Firoozi

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 $\epsilon$-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.

Suggested Citation

  • Hanchao Liu & Dena Firoozi, 2026. "LQ Mean Field Games with Common Noise in Hilbert Spaces: Small and Arbitrary Finite Time Horizons," Papers 2601.13493, arXiv.org.
    • Handle: RePEc:arx:papers:2601.13493
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