Recent progress in Carleman estimates for mean field games

This survey reviews recent advances in the application of Carleman estimates to Mean Field Games (MFG), a mathematical framework for modeling the dynamics of large populations of rational agents. The MFG system, governed by coupled nonlinear parabolic PDEs, presents challenges such as nonlinearity, nonlocal interactions, and bidirectional time evolution. Traditional analysis has required restrictive monotonicity assumptions to establish uniqueness and stability. A significant breakthrough was the introduction of Carleman estimates in the pioneering work of M. V. Klibanov and Y. Averboukh in 2023, which enabled the first Lipschitz and Hölder stability results for forward and inverse MFG problems. We highlight analytical developments in Coefficient Inverse Problems (CIPs), solved using the Bukhgeim–Klibanov method, and discuss the convexification method, which ensures globally convergent numerical solutions by embedding Carleman weight functions into optimization frameworks. These advances offer a unified analytical and computational strategy for addressing ill-posed problems in MFG, with potential applications to broader classes of PDE systems and data-driven models.