Holder stability and uniqueness for the mean field games system via Carleman estimates

We are concerned with the mathematical study of the mean field games system (MFGS). In the conventional setup, the MFGS is a system of two coupled nonlinear parabolic partial differential equation (PDE)s of the second order in a backward-forward manner, namely, one terminal and one initial condition are prescribed, respectively, for the value function and the population density . In this paper, we show that uniqueness of solutions to the MFGS can be guaranteed if, among all four possible terminal and initial conditions, either only two terminals or only two initial conditions are given. In both cases, Holder stability estimates are proven. This means that the accuracies of the solutions are estimated in terms of the given data. Moreover, these estimates readily imply uniqueness of corresponding problems for the MFGS. The main mathematical apparatus to establish those results is two new Carleman estimates, which may find application in other contexts associated with coupled parabolic PDEs.

Automated summary

This paper investigates the mathematical properties of the mean field games system (MFGS). It shows that the uniqueness of solutions to the MFGS can be guaranteed when only two terminal conditions or two initial conditions are given. Furthermore, Holder stability estimates are established. Two new Carleman estimates are introduced as the main mathematical tools, which may have applications in other contexts related to coupled parabolic PDEs.