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Martingales in self-similar growth-fragmentations and their applications
Post-edited
Authors : Bertoin, Jean (Author of the conference)
CIRM (Publisher )
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 (Publisher )
Abstract :
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.
MSC Codes : 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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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 23/06/2016 Conference Date : 07/06/2016 Subseries : Research talks arXiv category : Probability ; Mathematical Physics Mathematical Area(s) : Probability & Statistics Format : MP4 (.mp4) - HD Video Time : 01:00:16 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2016-06-07_Bertoin.mp4 
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Information on the Event
Event Title : Random trees and maps: probabilistic and combinatorial aspects / Arbres et cartes aléatoires : aspects probabilistes et combinatoiresEvent Organizers : Haas, Bénédicte ; Goldschmidt, Christina ; Miermont, Grégory Dates : 06/06/16 - 10/06/16 Event Year : 2016 Event URL : http://conferences.cirm-math.fr/1384.html
Citation Data
DOI : 10.24350/CIRM.V.18993003Cite this video as: 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 |
See Also
- [Multi angle] Recurrence of half plane maps / Author of the conference Angel, Omer.
- [Multi angle] Vanishing corrections for the position of an FKPP front / Author of the conference Berestycki, Julien.
- [Multi angle] Nesting statistics in the $O(n)$ loop model on random planar maps / Author of the conference Bouttier, Jérémie.
- [Multi angle] Vertex degrees in planar maps / Author of the conference Drmota, Michael.
Bibliography
- Bertoin, J., Budd, T., Curien, N., & Kortchemski, I. (2016). Martingales in self-similar growth-fragmentations and their connections with random planar maps.
