Global-in-time regularity via duality for congestion-penalized Mean Field Games

Abstract

After a brief introduction to one of the most typical problems in Mean Field Games, the local 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 .

Keywords:

• Mean Field Games
• convex duality
• Hamilton–Jacobi

Acknowledgements

The first author worked on this topic during his master studies at Ecole Polytechnique, funded by a Scholarship by Fondation Mathématique Jacques Hadamard, whose support is acknowledged. The second author acknowledges the support of the ANR project ISOTACE (ANR-12-MONU-0013) and of the iCODE project ‘Strategic Crowds’ funded by IDEX Paris-Saclay, and warmly thanks the organizers of International Conference on Stochastic Analysis and Applications, Hammamet, October 2015, for the opportunity to present the results and publish them.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Fondation Mathématique Jacques Hadamard; ANR Project ISOTACE [ANR-12-MONU-0013]; IDEX Paris-Saclay.