Geodesic
Displacement Convexity for First-Order Mean-Field Games
Minimax theory and its applications, Tome 3 (2018) no. 2
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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 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 q norms. This convexity gives bounds for the density of solutions of the planning problem and extends displacement convexity of q norms from optimal transport. Additionally, we prove the convexity of q norms for MFG with congestion.
Mots-clés : Mean field game, congestion, optimal transport, displacement convexity. 2010 Mathematics Subject Classification: 91A13, 35Q91, 26B25 
@article{MTA_2018_3_2_a3,
     author = {Diogo A. Gomes,Tommaso Seneci},
     title = {Displacement {Convexity} for {First-Order} {Mean-Field} {Games}},
     journal = {Minimax theory and its applications},
     year = {2018},
     volume = {3},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/MTA_2018_3_2_a3/}
}
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Diogo A. Gomes,Tommaso Seneci. Displacement Convexity for First-Order Mean-Field Games. Minimax theory and its applications, Tome 3 (2018) no. 2. http://geodesic.mathdoc.fr/item/MTA_2018_3_2_a3/