Optimal Transport, Wasserstein Gradient Flows and the JKO scheme

This course, given in September 2026 during a summer school on gradient flows in ICTS, Bangalore, has some common points with two courses that I gave in summer 2024, in Rome La Sapienza (Vito Volterra Meeting, June 24-28) and in Chania (Festum Pi festival, July 9-13).

Content and references

A general reference for the whole course is RoCha, written after the two summer schools in 2024.

  • 1) Introduction to optimal transport
    The Monge and the Kantorovich problem, and the dual of Kantorovich. Existence for Kantorovich and its dual. The Brenier theorem and its extensions. Wasserstein distances: triangle inequality, equivalence with the weak* convergence. First variation of the optimal transport costs.
    Roughly covered in section 3 of RoCha. Otherwise, see sections 1.1, 1.2, 1.3.1, 1.6.1, 5.1, 5.2 of OTAM.


  • 2) Introduction to gradient flows in Euclidean, metric, and Wasserstein spaces
    Implicit Euler, minimizing movements and the JKO. EDI definition in metric spaces. Guessing the evolution PDE from the optimality conditions in the JKO steps and the continuit equation. Variants corresponding to x'=Dh*(-DF(x)) for convex h (power-like or not).
    Roughly covered in Sections 2.1, 2.2, 3.3 of RoCha. Otherwise, see Sections 2, 3, 4.1-4.2-4.3 of the survey SGF (or Sections 8.1, 8.2 of OTAM) and, for the case in Wp, the paper CS1.


  • 3) Convergence of the JKO scheme for the Fokker-Planck equation
    Rigorous optimality conditions in the JKO steps. Geodesics in the Wasserstein space. Piecewise constant and piecewise geodesic interpolations. Identification of the limits.
    Very roughly covered in Section 4.1 of RoCha. Otherwise, see Section 4.4 of the survey SGF or Chapter 8 in the book OTAM, as well as section 5.3 of OTAM for the geodesics in the Wasserstein spaces.


  • 4) First estimates on the solutions of the JKO scheme and applications
    Upper and lower bounds on f'(ρ)+V. Five gradients inequality. BV estimates with arbitrry diffusion without potential. Fisher information estimates for the gradient flows of the entropy. Strong L2H2 convergence of the JKO scheme.
    The upper and lower bounds are not covered in RoCha, but you can find the upper one in Section 7.4.1 of OTAM, while the lower is Lemma 2.4 of the paper IPS. For the part based on five-gradients-inequality is covered in Sections 5 and 7 of RoCha; otherwise, see the paper DMSV, Cai, the last section in CS1, the paper DS and CS2 and finally ST.


  • 5) Other estimates on the solutions of the JKO scheme and applications
    (after some remarks on the previous lessons). The flow interchange technique for geodesically convex functionals. Lp estimates on the JKO scheme. Strong L2H1 convergence of the JKO scheme for nonlinear diffusion. A JKO-Bakry-Emery roof of the log-Sobolev inequality.
    Most of the lecture is covered in sections 4.2.1 and 6 of RoCha. For the strong convergence, see the beginning of Section 4 in ST. For the log-sobolev inequality, see the paper CC.


    • Links to references

    RoCha: the notes for the courses given in Rome and Chania in 2024, see here.
    SGF: a survey on gradient flows written in 2017, see here.
    OTAM: My book Optimal Transport for Applied Mathematicians (2015), see here or here for a non-official version.
    CS1: a paper with Thibault Caillet on the Wp case, see here. This paper generalizes previous results by Agueh and Otto. Except for the extensions to non-geodesically convex functionals (proven in CS1), all these papers are particular cases of a previous burt quite unknown paper by Figalli et al, see here.
    IPS: a paper with Mikaela Iacobelli and Francesco Patacchini on a certain gradient flow, where we prove the lower bounds, see here.
    DMSV: the paper with De Philippis, Mészáros and Velichkov on the five-gradients inequality and its application to BV estimates, see here.
    Cai: a paper by Thibault which generalizes the five-gradients inequality to other costs, see here.
    DS: a paper with Simone Di Marino where we prove Sobolev-like estimates (using Fisher-like quantities) on the JKO scheme for equations with linear diffusion, see here.
    CS2: a paper with Thibault Caillet on Fisher-like estimates for doubly nonlinear but 1-homogeneous equations. See here.
    ST: a paper with Gayrat Toshpulatov on the strong convergence of the JKO scheme for the linear Fokker-Planck equation. See here.
    MMS: a paper by Matthes (yes, Daniel), McCann and Savaré on 4th equations, where they first introduce the flow-interchange technique, see here.
    CC: a paper by Thibault Caillet and Fanch Coudreuse on the generalized log-Soboleven inequalities via the JKO scheme, see here.