English
Martingales in self-similar growth-fragmentations and their applications
Post-edited
Auteurs : Bertoin, Jean (Auteur de la Conférence)
CIRM (Editeur )
60G18 - Self-similar processes
60G44 - Martingales with continuous parameter
60G50 - Sums of independent random variables; random walks
60G51 - Processes with independent increments; Lévy processes
60J75 - Jump processes
CIRM (Editeur )
Résumé :
This talk is based on a work jointly with Timothy Budd (Copenhagen), Nicolas Curien (Orsay) and Igor Kortchemski (Ecole Polytechnique).
Consider a self-similar Markov process $X$ on $[0,\infty)$ which converges at infinity a.s. We interpret $X(t)$ as the size of a typical cell at time $t$, and each negative jump as a birth event. More precisely, if ${\Delta}X(s) = -y < 0$, then $s$ is the birth at time of a daughter cell with size $y$ which then evolves independently and according to the same dynamics. In turn, daughter cells give birth to granddaughter cells each time they make a negative jump, and so on.
The genealogical structure of the cell population can be described in terms of a branching random walk, and this gives rise to remarkable martingales. We analyze traces of these mar- tingales in physical time, and point at some applications for self-similar growth-fragmentation processes and for planar random maps.
Codes MSC : Consider a self-similar Markov process $X$ on $[0,\infty)$ which converges at infinity a.s. We interpret $X(t)$ as the size of a typical cell at time $t$, and each negative jump as a birth event. More precisely, if ${\Delta}X(s) = -y < 0$, then $s$ is the birth at time of a daughter cell with size $y$ which then evolves independently and according to the same dynamics. In turn, daughter cells give birth to granddaughter cells each time they make a negative jump, and so on.
The genealogical structure of the cell population can be described in terms of a branching random walk, and this gives rise to remarkable martingales. We analyze traces of these mar- tingales in physical time, and point at some applications for self-similar growth-fragmentation processes and for planar random maps.
60G18 - Self-similar processes
60G44 - Martingales with continuous parameter
60G50 - Sums of independent random variables; random walks
60G51 - Processes with independent increments; Lévy processes
60J75 - Jump processes
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 23/06/2016 Date de captation : 07/06/2016 Sous collection : Research talks arXiv category : Probability ; Mathematical Physics Domaine : Probability & Statistics Format : MP4 (.mp4) - HD Durée : 01:00:16 Audience : Researchers Download : https://videos.cirm-math.fr/2016-06-07_Bertoin.mp4 
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Informations sur la Rencontre
Nom de la rencontre : Random trees and maps: probabilistic and combinatorial aspects / Arbres et cartes aléatoires : aspects probabilistes et combinatoiresOrganisateurs de la rencontre : Haas, Bénédicte ; Goldschmidt, Christina ; Miermont, Grégory Dates : 06/06/16 - 10/06/16 Année de la rencontre : 2016 URL Congrès : http://conferences.cirm-math.fr/1384.html
Données de citation
DOI : 10.24350/CIRM.V.18993003Citer cette vidéo: Bertoin, Jean (2016). Martingales in self-similar growth-fragmentations and their applications. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18993003 URI : http://dx.doi.org/10.24350/CIRM.V.18993003 |
Voir aussi
- [Multi angle] Recurrence of half plane maps / Auteur de la Conférence Angel, Omer.
- [Multi angle] Vanishing corrections for the position of an FKPP front / Auteur de la Conférence Berestycki, Julien.
- [Multi angle] Nesting statistics in the $O(n)$ loop model on random planar maps / Auteur de la Conférence Bouttier, Jérémie.
- [Multi angle] Vertex degrees in planar maps / Auteur de la Conférence Drmota, Michael.
Bibliographie
- Bertoin, J., Budd, T., Curien, N., & Kortchemski, I. (2016). Martingales in self-similar growth-fragmentations and their connections with random planar maps.
