Skip to main content
Log in

Growth-fragmentation process embedded in a planar Brownian excursion

  • Published:
Probability Theory and Related Fields Aims and scope Submit manuscript

Abstract

The aim of this paper is to present a self-similar growth-fragmentation process linked to a Brownian excursion in the upper half-plane \({\mathbb {H}}\), obtained by cutting the excursion at horizontal levels. We prove that the associated growth-fragmentation is related to one of the growth-fragmentation processes introduced by Bertoin, Budd, Curien and Kortchemski in (Bertoin et al. Probab Theory Relat Field 172:663–724, 2018).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Aïdékon, E., Hu, Y., Shi, Z.: Points of infinite multiplicity of planar Brownian motion: measures and local times. Ann. Probab. 48(4), 1785–1825 (2020)

    Article  MathSciNet  Google Scholar 

  2. Bertoin, J.: Lévy processes.In: Cambridge Tracts in Mathematics, 121. Cambridge University Press. Cambridge (1996)

  3. Bertoin, J.: Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38(3), 319–340 (2002)

    Article  MathSciNet  Google Scholar 

  4. Bertoin, J.: Markovian growth-fragmentation processes. Bernoulli 23, 1082–1101 (2017)

    Article  MathSciNet  Google Scholar 

  5. Bertoin, J., Budd, T., Curien, N., Kortchemski, I.: Martingales in self-similar growth-fragmentations and their connections with random planar maps. Probab. Theory Relat. Fields 172, 663–724 (2018)

    Article  MathSciNet  Google Scholar 

  6. Bertoin, J., Curien, N., Kortchemski, I.: On conditioning a self-similar growth-fragmentation by its intrinsic area. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 57(2), 1136–1156 (2021)

  7. Biggins, J.D., Kyprianou, A.E.: Measure change in multitype branching. Adv. Appl. Probab. 36, 544–581 (2004)

    Article  MathSciNet  Google Scholar 

  8. Blumenthal, R.M., Getoor, R.K., Ray, D.B.: On the distribution of first hits for the symmetric stable processes. Trans. Am. Math. Soc 99(3), 540–554 (1961)

    MathSciNet  MATH  Google Scholar 

  9. Caballero, M.E., Chaumont, L.: Conditioned stable Lévy process and the Lamperti representation. J. Appl. Probab. 43(4), 967–983 (2006)

    Article  MathSciNet  Google Scholar 

  10. Chaumont, Loïc., Pantí, Henry, Rivero, Víctor.: The Lamperti representation of real-valued self-similar Markov processes. Bernoulli 19(5B), 2494–2523 (2013)

    Article  MathSciNet  Google Scholar 

  11. Chen, L., Curien, N., Maillard, P.: The perimeter cascade in critical Boltzmann quadrangulations decorated by an \(O(n)\) loop model. Ann. de l’Inst. Henri Poincare D7(4), 535-84

  12. Da Silva, W.: Self-similar signed growth-fragmentations. arXiv:2101.02582 (2021)

  13. Kuznetsov, A., Pardo, J.C.: Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Acta Appl. Math. 123, 113–139 (2013)

    Article  MathSciNet  Google Scholar 

  14. Lawler, G.F.: Conformally invariant processes in the plane. Mathematical Surveys and Monographs, 114. American Mathematical Society, Providence, RI (2005)

  15. Le Gall, J.F., Miermont, G.: Scaling limits of random planar maps with large faces. Ann. Probab. 39(1), 1–69 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Le Gall, J.-F., Riera, A.: Growth-fragmentation processes in Brownian motion indexed by the Brownian tree. Ann. Probab. 48(4), 1742–1784 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Revuz, D., Yor, M.: Continuous martingales and Brownian motion.In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer-Verlag, Berlin (1999)

  18. Shi, Z.: Branching random walks of Lecture Notes in Mathematics, Springer, Cham. Lecture notes from the 42nd probability summer school held in Saint flour vol. 2151 2012, Ecole d’Eté de Probabilités de Saint-Flour (2015)

Download references

Acknowledgements

We are grateful to Jean Bertoin and Bastien Mallein for stimulating discussions, and to Juan Carlos Pardo for a number of helpful discussions regarding self-similar processes. After a first version of this article appeared online, Nicolas Curien pointed to us the connection with random planar maps, and the link between the duration of the excursion and the area of the map. We warmly thank him for his explanations. We also learnt that Timothy Budd in an unpublished note had already predicted the link between growth-fragmentations of [5] and planar excursions. Finally, we thank two anonymous referees for their careful reading and valuable comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to William Da Silva.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aïdékon, E., Da Silva, W. Growth-fragmentation process embedded in a planar Brownian excursion. Probab. Theory Relat. Fields 183, 125–166 (2022). https://doi.org/10.1007/s00440-022-01119-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-022-01119-y

Keywords

Mathematics Subject Classification

Navigation