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

Course at Universitat Autònoma de Barcelona, in the research program Geometric function theory in fluid mechanics.

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 W2; geodesics in W2; 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 Lp estimates and L2H1 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.