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H 1 The matching problem: connections to the Gaussian free field via large-scale linearization of the Monge-Ampere equation

Auteurs : Otto, Felix (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : The optimal transport between a random atomic measure described by the Poisson point process and the Lebesgue measure in d-dimensional space has received attention in diverse communities. Heuristics suggest that on large scales, the displacement potential, which is a solution of the highly nonlinear Monge-Ampere equation with a rough right hand side, behaves like the solution of its linearization, the Poisson equation driven by white noise. Most interesting is the case of dimension d=2, when the displacement inherits the logarithmic divergence of the Gaussian free field. For a large torus, this has been made rigorous on the macroscopic level (i.e. on the size of the torus) by recent work of Ambrosio.et.al.
    We show that this is also true on the microscopic level (i.e. on the scale of the point process). The argument relies on a new and purely variational approach to the (Schauder) regularity theory for the Monge-Ampere equation, which allows for a rough right hand side, and which amounts to a quantitative linearization on all (intermediate) scales. This deterministic approach allows to feed in the existing stochastic estimates. This is joint work with M.Goldman and M.Huesmann.

    Keywords : optimal transportation, matching

    Codes MSC :
    60G55 - Point processes
    35J96 - Elliptic Monge-Ampère equations

      Informations sur la Vidéo

      Réalisateur : Recanzone, Luca
      Langue : Anglais
      Date de publication : 04/11/2019
      Date de captation : 15/10/2019
      Collection : Research talks
      Format : MP4
      Durée : 00:58:31
      Domaine : PDE ; Probability & Statistics
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2019-10-15_Otto.mp4

    Informations sur la rencontre

    Nom de la rencontre : PDE/Probability Interactions: Particle Systems, Hyperbolic Conservation Laws / Interactions EDP/Probabilités : systèmes de particules, lois de conservation hyperboliques
    Organisateurs de la rencontre : Caputo, Pietro ; Fathi, Max ; Guillin, Arnaud ; Reygner, Julien
    Dates : 14/10/2019 - 18/10/2019
    Année de la rencontre : 2019
    URL Congrès : https://conferences.cirm-math.fr/2083.html

    Citation Data

    DOI : 10.24350/CIRM.V.19570403
    Cite this video as: Otto, Felix (2019). The matching problem: connections to the Gaussian free field via large-scale linearization of the Monge-Ampere equation. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19570403
    URI : http://dx.doi.org/10.24350/CIRM.V.19570403

    Voir aussi


    1. GOLDMAN, Michael, HUESMANN, Martin, et OTTO, Felix. A large-scale regularity theory for the Monge-Ampère equation with rough data and application to the optimal matching problem. arXiv preprint arXiv:1808.09250, 2018. - https://arxiv.org/abs/1808.09250