Variational Mean Field Games and Optimal Transport

Course at GSSI, Workshop/School on stochastic PDEs, mean-field games, and biology.

Program

3 classes of 1h15 each, tentatively organized as follows.

  • Lecture 1 (4/9) An overview of optimal transport.
    Monge and Kantorovich problems; Kantorovich duality; Brenier's theorem and existence of optimal maps; Wasserstein distances and spaces; some notions about curves and geodesics in metric spaces; AC curves in Wp; geodesics in Wp; the Benamou-Brenier dynamical problem; duality for Benamou-Brenier.
  • Lecture 2 (5/9) Regularity via duality.
    First-order variational MFG. Eulerian and Lagrangian formulations. Equivalences between optimality and equilibrium: formal derivation of the MFG system in the Eulerian framewrok, rigorous derivation of the optimality of a.e. trajectory in the Lagrangian one. Need for regularity and/or summability. Methods for Sobolev regularity based on duality.
  • Lecture 3 (6/9) Regularity via OT and time-discretization.
    Time-discretization of minimal action problems. Flow interchange and geodesic convexity. L∞ estimates.

    • References:

    For a general introduction to variational first-order MFG with local coupling, see this survey paper.

    All the material covered in Lecture 1 is contained in the recent book Optimal Transport for Applied Mathematicians (OTAM; see here or here for a non-official version), and in particular in Sections 1.1, 1.2, 1.3, 5.1, 5.2, 5.3, 5.4, 6.1.

    For Lecture 2, see this paper on regularity via duality for MFG. For general ideas about regularity via duality, you can also look at this other paper, and for the proof of the equilibrium as an optimality condition in the Lagrangian model, at Section 7 of this other paper (which deals with a more delicate situation, that of density constraints).

    For Lecture 3, see this other paper. For the first-order variation of the Wasserstein distance, see Section 7.2.2 of OTAM, and for geodesic convexity see Section 7.3.2.

    Of course, there are more classical references for optimal transport: the first book by Cédric Villani Topics in Optimal Transportation (Am. Math. Soc., GSM, 2003) on the general theory, and Gradient Flows in Metric Spaces and in the Space of Probabiliy Measures, by Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré (Birkhäuser, 2005).