On the long-time behavior of mean field game systems with a common noise

Abstract: In this paper, we study the long-time behavior of mean field game (MFG) systems influenced by a common noise. While classical results establish the convergence of deterministic MFG towards stationary solutions under suitable monotonicity conditions, the introduction of a common stochastic perturbation significantly complicates the analysis. We consider a standard MFG model with infinitely many players whose dynamics are subject to both idiosyncratic and common noise. The central goal is to characterize the asymptotic properties as the horizon goes to infinity. By employing quantitative methods that replace classical compactness arguments unavailable in the stochastic context, we prove that solutions exhibit exponential convergence toward a stationary regime. Specifically, we identify a deterministic ergodic constant and demonstrate the existence of stationary random processes capturing the limiting behavior. Further, we establish almost sure long-time results thanks to a detailed analysis of the ergodic master equation, which is the long-time limit of the master equation. Our results extend known deterministic convergence phenomena to the stochastic setting, relying on novel backward stochastic PDE estimates.