Electronic Journal of Probability

Profile of a self-similar growth-fragmentation

François Gaston Ged

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Abstract

A self-similar growth-fragmentation describes the evolution of particles that grow and split as time passes. Its genealogy yields a self-similar continuum tree endowed with an intrinsic measure. Extending results of Haas [13] for pure fragmentations, we relate the existence of an absolutely continuous profile to a simple condition in terms of the index of self-similarity and the so-called cumulant of the growth-fragmentation. When absolutely continuous, we approximate the profile by a function of the small fragments, and compute the Hausdorff dimension in the singular case. Applications to Boltzmann random planar maps are emphasized, exploiting recently established connections between growth-fragmentations and random maps [1, 5].

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 7, 21 pp.

Dates
Received: 12 April 2018
Accepted: 3 December 2018
First available in Project Euclid: 15 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1550199785

Digital Object Identifier
doi:10.1214/18-EJP253

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60G18: Self-similar processes 60G30: Continuity and singularity of induced measures

Keywords
self-similar growth-fragmentations intrinsic area profile of a tree

Rights
Creative Commons Attribution 4.0 International License.

Citation

Ged, François Gaston. Profile of a self-similar growth-fragmentation. Electron. J. Probab. 24 (2019), paper no. 7, 21 pp. doi:10.1214/18-EJP253. https://projecteuclid.org/euclid.ejp/1550199785


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References

  • [1] J. Bertoin, T. Budd, N. Curien, and I. Kortchemski, Martingales in self-similar growth-fragmentations and their connections with random planar maps, To appear in Probability Theory and Related Fields (2016).
  • [2] Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996.
  • [3] Jean Bertoin, On small masses in self-similar fragmentations, Stochastic Process. Appl. 109 (2004), no. 1, 13–22.
  • [4] Jean Bertoin, Markovian growth-fragmentation processes, Bernoulli 23 (2017), no. 2, 1082–1101.
  • [5] Jean Bertoin, Nicolas Curien, and Igor Kortchemski, Random planar maps & growth-fragmentations, To appear in Ann. Probab.
  • [6] Jean Bertoin, Alexander Lindner, and Ross Maller, On continuity properties of the law of integrals of Lévy processes, Séminaire de probabilités XLI, Lecture Notes in Math., vol. 1934, Springer, Berlin, 2008, pp. 137–159.
  • [7] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987.
  • [8] S. Bochner and K. Chandrasekharan, Fourier Transforms, Annals of Mathematics Studies, no. 19, Princeton University Press, Princeton, N. J.; Oxford University Press, London, 1949.
  • [9] Philippe Carmona, Frédérique Petit, and Marc Yor, On the distribution and asymptotic results for exponential functionals of Lévy processes, Exponential functionals and principal values related to Brownian motion, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 1997, pp. 73–130.
  • [10] Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel. Chapitres V à VIII, revised ed., Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], vol. 1385, Hermann, Paris, 1980, Théorie des martingales. [Martingale theory].
  • [11] Jean-François Delmas, Fragmentation at height associated with Lévy processes, Stochastic Process. Appl. 117 (2007), no. 3, 297–311.
  • [12] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986.
  • [13] Bénédicte Haas, Regularity of formation of dust in self-similar fragmentations, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 4, 411–438.
  • [14] Bénédicte Haas and Grégory Miermont, The genealogy of self-similar fragmentations with negative index as a continuum random tree, Electron. J. Probab. 9 (2004), no. 4, 57–97.
  • [15] Andreas E. Kyprianou, Fluctuations of Lévy processes with applications, second ed., Universitext, Springer, Heidelberg, 2014, Introductory lectures.
  • [16] John Lamperti, Semi-stable Markov processes. I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 205–225.
  • [17] Elliott H. Lieb and Michael Loss, Analysis, second ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001.
  • [18] Quansheng Liu, On generalized multiplicative cascades, Stochastic Process. Appl. 86 (2000), no. 2, 263–286.
  • [19] Pierre Patie and Mladen Savov, Bernstein-gamma functions and exponential functionals of Lévy processes, arXiv:1604.05960 (2016).
  • [20] Franz Rembart and Matthias Winkel, Recursive construction of continuum random trees, To appear in Annals of Probability (2016).
  • [21] Quan Shi, Growth-fragmentation processes and bifurcators, Electron. J. Probab. 22 (2017), 25 pp.