Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limit and ageing for branching random walk in Pareto environment

Marcel Ortgiese and Matthew I. Roberts

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show that the system of particles, rescaled in an appropriate way, converges in distribution to a scaling limit that is interesting in its own right. We describe the limit object as a growing collection of “lilypads” built on a Poisson point process in $\mathbb{R}^{d}$. As an application of our main theorem, we show that the maximizer of the system displays the ageing property.

Résumé

Nous considérons une marche aléatoire branchante sur un réseau, où les taux de branchement sont donnés par un potentiel aléatoire i.i.d. suivant des lois de Pareto. Nous montrons que le système de particules, renormalisé d’une façon idoine, converge en loi vers une limite d’échelle intéressante en elle-même. Nous décrivons l’objet limite comme une collection croissante de « nénuphars » construits à partir d’un processus de Poisson dans $\mathbb{R}^{d}$. Comme application de notre théorème principal, nous montrons que le maximiseur du système possède la propriété de vieillissement.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1291-1313.

Dates
Received: 29 April 2016
Revised: 15 March 2017
Accepted: 20 April 2017
First available in Project Euclid: 11 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1531296020

Digital Object Identifier
doi:10.1214/17-AIHP839

Mathematical Reviews number (MathSciNet)
MR3825882

Zentralblatt MATH identifier
06976076

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching random walk Random environment Parabolic Anderson model Intermittency

Citation

Ortgiese, Marcel; Roberts, Matthew I. Scaling limit and ageing for branching random walk in Pareto environment. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1291--1313. doi:10.1214/17-AIHP839. https://projecteuclid.org/euclid.aihp/1531296020


Export citation

References

  • [1] S. Albeverio, L. V. Bogachev, S. A. Molchanov and E. B. Yarovaya. Annealed moment Lyapunov exponents for a branching random walk in a homogeneous random branching environment. Markov Process. Related Fields 6 (4) (2000) 473–516.
  • [2] A. Baddeley. Spatial point processes and their applications. In Stochastic Geometry 1–75. Lecture Notes in Math. 1892. Springer, Berlin, 2007.
  • [3] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York, 1999. A Wiley-Interscience Publication.
  • [4] M. Biskup, W. König and R. S. dos Santos. Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails. Probab. Theory Related Fields 171 (1–2) (2018). 251–331.
  • [5] F. Comets and S. Popov. Shape and local growth for multidimensional branching random walks in random environment. ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007) 273–299.
  • [6] S. N. Ethier and T. G. Kurtz. Markov Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York, 1986. Characterization and convergence.
  • [7] A. Fiodorov and S. Muirhead. Complete localisation and exponential shape of the parabolic Anderson model with Weibull potential field. Electron. J. Probab. 19 (58) (2014) 27.
  • [8] O. Gün, W. König and O. Sekulović. Moment asymptotics for branching random walks in random environment. Electron. J. Probab. 18 (63) (2013) 18.
  • [9] J. Gärtner and S. A. Molchanov. Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm. Math. Phys. 132 (3) (1990) 613–655.
  • [10] R. van der Hofstad, P. Mörters and N. Sidorova. Weak and almost sure limits for the parabolic Anderson model with heavy-tailed potentials. Ann. Appl. Probab. 18 (6) (2008) 2450–2494.
  • [11] W. König, H. Lacoin, P. Mörters and N. Sidorova. A two cities theorem for the parabolic Anderson model. Ann. Probab. 37 (2009) 347–392.
  • [12] P. Mörters, M. Ortgiese and N. Sidorova. Ageing in the parabolic Anderson model. Ann. Inst. Henri Poincaré Probab. Stat. 47 (4) (2011) 969–1000.
  • [13] M. Ortgiese and M. I. Roberts. Intermittency for branching random walk in Pareto environment. Ann. Probab. 44 (3) (2016) 2198–2263.
  • [14] M. Ortgiese and M. I. Roberts. One-point localisation for branching random walk in Pareto environment. Electron. J. Probab. 22 (6) (2017).
  • [15] S. I. Resnick. Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2008. Reprint of the 1987 original.
  • [16] N. Sidorova and A. Twarowski. Localisation and ageing in the parabolic Anderson model with Weibull potential. Ann. Probab. 42 (4) (2014) 1666–1698.