Inserted: 30 mar 2016
Last Updated: 16 jan 2017
Proceedings of the International Conference on Stochastic Analysis and Applications, 2015, Hammamet.
After a brief introduction to one of the most typical problems in Mean Field Games, the congestion case (where agents pay a cost depending on the density of the regions they visit), and to its variational structure, we consider the question of the regularity of the optimal solutions. A duality argument, used for the first time in a paper by Y. Brenier on incompressible fluid mechanics, and recently applied to MFG with density constraints, allows to easily get some Sobolev regularity, locally in space and time. In the paper we prove that a careful analysis of the behaviour close to the final time allows to extend the same result including $t=T$.
Keywords: mean field games, convex duality, Wasserstein space, Benamou-Brenier