On The Mean Field Games System With the Lateral Cauchy Data via Carleman Estimates

We are concerned with the quantitative study of the second order Mean Field Game (MFG) system in a bounded domain with the lateral Cauchy data being prescribed. That is, both Dirichlet and Neumann boundary data of the MFG solutions are given. We derive a sharp H\"older stability estimate in quantifying the difference of the MFG solutions in terms of the corresponding difference of their lateral Cauchy data. This stability estimate implies uniqueness. That is, the lateral Cauchy data uniquely determine the MFG solutions. Some applications of practical interest are discussed. The main technical developments are two new Carleman estimates for the MFG system.