# First-order and second-order evolution in the Wasserstein space

### Program

5 classes of 50' each, tentatively organized as follows.

* Lecture 1 (2/7)* ** Preliminaries on optimal transport.**

Monge and Kantorovich problems; Kantorovich duality; Brenier's theorem
and existence of optimal maps; Wasserstein distances and spaces.
* Lecture 2 (3/7)*** Curves in metric spaces and in the Wasserstein
space.**

Some notions
about curves and geodesics in metric spaces; AC curves in
W_{2}; geodesics in W_{2}; the Benamou-Brenier
dynamical problem...
* Lecture 3 (4/7)* ** Gradient flows.**

Gradient flows in Euclidean and metric spaces. Time
discretization. The JKO scheme in the Wasserstein space. First
variations of various functionals on the space of probabilities. The
Fokker-Planck example.
* Lecture 4 (5/7)*** Flow interchange and stronger
estimates for non-linear equations.**

Digression on geodesically convex functionals: internal,
potential, and interaction energies (McCann's condition). Presentation
of the porous medium equation as a gradient flow. Uniform L^{p}
estimates and L^{2}H^{1} estimates via the flow
interchange with power functionals
* Lecture 5 (6/7)*** Second-order problems and MFG.**

Equilibrium problems in variational Mean Field Games. Hamilton-Jacobi
equation and duality. Optimal curves in the Wasserstein space: time
discretization and flow interchange estimates.
### References:

Most of the material covered in the lectures is contained in the recent book
*Optimal Transport for Applied Mathematicians* (OTAM; see here or here for a non-official version).

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).

**Precise references lecture by lecture**

Lecture 1: OTAM, Sections 1.1, 1.2, 1.3, 5.1, 5.2

Lecture 2: OTAM, Sections 5.3, 5.4, 6.1

Lecture 3: OTAM, Sections 7.2, 8.1, 8.2, 8.3. You can also see this survey.

Lecture 4: OTAM, Sections 7.3, 8.4. The flow interchange technique has
been introduced in this paper.

Lecture 5: OTAM, Section 8.4; for an introduction to variational MFG,
see this survey. The
application of the flow interchange to MFG is in this paper.