Displacement convexity for first-order mean-field games

Here, we consider the planning problem for first-order mean-field games (MFG). When there is no coupling between players, MFG degenerate into optimal transport problems. Displacement convexity is a fundamental tool in optimal transport that often reveals hidden convexity of functionals and, thus, has numerous applications in the calculus of variations. We explore the similarities between the Benamou-Brenier formulation of optimal transport and MFG to extend displacement convexity methods from to MFG. In particular, we identify a class of functions, that depend on solutions of MFG, that are convex in time and, thus, obtain new a priori bounds for solutions of MFG. A remarkable consequence is the log-convexity of $L^q$ norms. This convexity gives bounds for the density of solutions of the planning problem and extends displacement convexity of $L^q$ norms from optimal transport. Additionally, we prove the convexity of $L^q$ norms for MFG with congestion.