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H 1 Vanishing corrections for the position of an FKPP front

Auteurs : Berestycki, Julien (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : The celebrated Fisher-Kolmogorov-Petrovsky-Piscounof equation (FKPP) in one dimension for
    $h:\mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}$ is:

    $\partial_th = \partial{_x^2}h + h - h^2, h(x, 0) = h_0(x)$.

    This equation is a natural description of a reaction-diffusion model (Fisher 1937, Kolmogorov et al. 1937, Aronson 1978). It is also related to branching Brownian motion: for the Heaviside initial condition $h_0 (x) = 1{_x<0}$ , $h(x, t)$ is the probability that the rightmost particle at time t in a branching Brownian motion (BBM) is to the right of $x$.
    One of the beauty of this equation is that for initial conditions that decrease sufficiently fast, a front develops, i.e. there exists a centring term $m(t)$ and an asymptotic shape $\omega(x)$ such that

    $\lim_{t \to \infty} h(m(t) + x,t) = \omega(x) \in (0, 1).$

    Since the original paper of Kolmogorov et al., the position of the front $m(t)$ has been studied intensely, in particular by Bramson. In this talk, I will present some recent results concerning a prediction of Ebert and van Saarloos about the vanishing corrections of this position.
    Based on a joint work with E. Brunet.

    Codes MSC :
    35K57 - Reaction-diffusion equations
    60J80 - Branching processes (Galton-Watson, birth-and-death, etc.)

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 23/06/2016
      Date de captation : 09/06/2016
      Collection : Research talks ; Partial Differential Equations ; Probability and Statistics
      Format : MP4
      Durée : 00:55:44
      Domaine : Probability & Statistics ; PDE
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2016-06-09_Berestycki.mp4

    Informations sur la rencontre

    Nom de la rencontre : Random trees and maps: probabilistic and combinatorial aspects / Arbres et cartes aléatoires : aspects probabilistes et combinatoires
    Organisateurs de la rencontre : Haas, Bénédicte ; Goldschmidt, Christina ; Miermont, Grégory
    Dates : 06/06/2016 - 10/06/2016
    Année de la rencontre : 2016
    URL Congrès : http://conferences.cirm-math.fr/1384.html

    Citation Data

    DOI : 10.24350/CIRM.V.18996203
    Cite this video as: Berestycki, Julien (2016). Vanishing corrections for the position of an FKPP front. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18996203
    URI : http://dx.doi.org/10.24350/CIRM.V.18996203


    Voir aussi

    Bibliographie

    1. Berestycki, J., Brunet, E., Harris, S.C., & Roberts, Matthew I. (2015). Vanishing corrections for the position in a linear model of FKPP fronts. - https://arxiv.org/abs/1510.03329

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