Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-28T03:50:24.362Z Has data issue: false hasContentIssue false
  • English
  • Français

A growth-fragmentation model connected to the ricocheted stable process

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  08 November 2022

Alexander R. Watson Open the ORCID record for Alexander R. Watson [Opens in a new window]
Alexander R. Watson*
Affiliation:
University College London
*
*Postal address: Gower Street, WC1E 6BT, UK. Email: alexander.watson@ucl.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Growth-fragmentation processes describe the evolution of systems in which cells grow slowly and fragment suddenly. Despite originating as a way to describe biological phenomena, they have recently been found to describe the lengths of certain curves in statistical physics models. In this note, we describe a new growth-fragmentation process connected to random planar maps with faces of large degree, having as a key ingredient the ricocheted stable process recently discovered by Budd. The process has applications to the excursions of planar Brownian motion and Liouville quantum gravity.

Keywords

random planar mapsstatistical physicsLévy processstable processself-similar Markov process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G18: Self-similar processes
Secondary: 60G52: Stable processes 60G51: Processes with independent increments; L'evy processes
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 493 - 503
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aïdékon, E. and Da Silva, W. (2020). Growth-fragmentation process embedded in a planar Brownian excursion. Prob. Theory Relat. Fields 183, 125–166Google Scholar
Bertoin, J. (2002). Self-similar fragmentations. Ann. Inst. H. Poincaré Prob. Statist. 38, 319–340.CrossRefGoogle Scholar
Bertoin, J. (2006). Random Fragmentation and Coagulation Processes (Cambridge Studies Adv. Math. 102). Cambridge University Press.Google Scholar
Bertoin, J. (2017). Markovian growth-fragmentation processes. Bernoulli 23, 1082–1101.CrossRefGoogle Scholar
Bertoin, J. and Mallein, B. (2019). Infinitely ramified point measures and branching Lévy processes. Ann. Prob. 47, 1619–1652.CrossRefGoogle Scholar
Bertoin, J., Budd, T., Curien, N. and Kortchemski, I. (2018). Martingales in self-similar growth-fragmentations and their connections with random planar maps. Prob. Theory Relat. Fields 172, 663–724.CrossRefGoogle Scholar
Bertoin, J., Curien, N. and Kortchemski, I. (2018). Random planar maps and growth-fragmentations. Ann. Prob. 46, 207–260.CrossRefGoogle Scholar
Blumenthal, R. M., Getoor, R. K. and Ray, D. B. (1961). On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99, 540–554.Google Scholar
Budd, T. (2018). The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen). Preprint, arXiv:1809.02012v1 [math.PR].Google Scholar
Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2011). Explicit identities for Lévy processes associated to symmetric stable processes. Bernoulli 17, 34–59.CrossRefGoogle Scholar
Chaumont, L. and Rivero, V. (2007). On some transformations between positive self-similar Markov processes. Stoch. Process. Appl. 117, 1889–1909.CrossRefGoogle Scholar
Da Silva, W. (2021). Self-similar signed growth-fragmentations. Preprint, arXiv:2101.02582v1 [math.PR].Google Scholar
Kuznetsov, A. and Pardo, J. C. (2013). Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Acta Appl. Math. 123, 113–139.CrossRefGoogle Scholar
Kuznetsov, A., Kyprianou, A. E., Pardo, J. C. and Watson, A. R. (2014). The hitting time of zero for a stable process. Electron. J. Prob. 19, 1–26.CrossRefGoogle Scholar
Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2n edn. Springer, New York.Google Scholar
Kyprianou, A. E., Pardo, J. C. and Watson, A. R. (2014). Hitting distributions of $\alpha$ -stable processes via path censoring and self-similarity. Ann. Prob. 42, 398–430.Google Scholar
Kyprianou, A. E., Pardo, J. C. and Vidmar, M. (2020). Double hypergeometric Lévy processes and self-similarity. Preprint, arXiv:1904.06143v2 [math.PR].Google Scholar
Kyprianou, A. E., Rivero, V. M. and Satitkanitkul, W. (2019). Conditioned real self-similar Markov processes. Stoch. Process. Appl. 129, 954–977.CrossRefGoogle Scholar
Miller, J., Sheffield, S. and Werner, W. (2021). Simple conformal loop ensembles on Liouville quantum gravity. Preprint, arXiv:2002.05698v4 [math.PR].Google Scholar
Rogozin, B. A. (1972). The distribution of the first hit for stable and asymptotically stable walks on an interval. Theory Prob. Appl. 17, 332–338.CrossRefGoogle Scholar
Shi, Q. (2017). Growth-fragmentation processes and bifurcators. Electron. J. Prob. 22, 1–25.CrossRefGoogle Scholar