# Variational Mean Field Games and Optimal Transport

### 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
W_{p}; geodesics in W_{p}; 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).