| Abstract: |
| We study the long-time behavior of mean field game (MFG) systems driven by common noise, providing a natural yet unexplored extension of the deterministic MFG theory. In the deterministic setting, classical results establish convergence toward stationary solutions under suitable monotonicity assumptions. The presence of common stochastic perturbations, however, makes the analysis substantially more delicate. We consider a standard MFG model with infinitely many players whose dynamics are affected by both idiosyncratic and common noise, and we investigate its asymptotic behavior as the time horizon tends to infinity. Using quantitative arguments that replace the compactness methods available in the deterministic framework, we prove exponential convergence of the solutions toward a stationary regime. More precisely, we identify a deterministic ergodic constant and show the existence of stationary random processes describing the limiting behavior. We also establish almost sure long-time convergence through a detailed analysis of the ergodic master equation, which governs the asymptotic behavior of the master equation. |
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