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.
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
about curves and geodesics in metric spaces; AC curves in
W2; geodesics in W2; the Benamou-Brenier
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
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.
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,
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.