Electronic Journal of Probability

One-point localization for branching random walk in Pareto environment

Marcel Ortgiese and Matthew I. Roberts

Full-text: Open access

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 a very strong form of intermittency, where with high probability most of the mass of the system is concentrated in a single site with high potential. The analogous one-point localization is already known for the parabolic Anderson model, which describes the expected number of particles in the same system. In our case, we rely on very fine estimates for the behaviour of particles near a good point. This complements our earlier results that in the rescaled picture most of the mass is concentrated on a small island.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 6, 20 pp.

Dates
Received: 29 April 2016
Accepted: 22 December 2016
First available in Project Euclid: 17 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1484622023

Digital Object Identifier
doi:10.1214/16-EJP22

Mathematical Reviews number (MathSciNet)
MR3613699

Zentralblatt MATH identifier
1357.60093

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 intermittency localization

Rights
Creative Commons Attribution 4.0 International License.

Citation

Ortgiese, Marcel; Roberts, Matthew I. One-point localization for branching random walk in Pareto environment. Electron. J. Probab. 22 (2017), paper no. 6, 20 pp. doi:10.1214/16-EJP22. https://projecteuclid.org/euclid.ejp/1484622023


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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):473–516, 2000.
  • [2] J. Gärtner and W. König. The parabolic Anderson model. In Interacting stochastic systems, pages 153–179. Springer, Berlin, 2005.
  • [3] O. Gün, W. König, and O. Sekulović. Moment asymptotics for branching random walks in random environment. Electron. J. Probab., 18(63):1–18, 2013.
  • [4] J. Gärtner and S. A. Molchanov. Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm. Math. Phys., 132(3):613–655, 1990.
  • [5] 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):2450–2494, 2008.
  • [6] S.C. Harris and M. I. Roberts. The many-to-few lemma and multiple spines. To appear in Ann. Inst. Henri Poincaré. Preprint arXiv:1106.4761, 2016.
  • [7] W. König, H. Lacoin, P. Mörters, and N. Sidorova. A two cities theorem for the parabolic Anderson model. Ann. Probab., 37(1):347–392, 2009.
  • [8] W. König. The parabolic Anderson model. Random walk in random potential. Birkhäuser Basel, Basel, 2016.
  • [9] P. Mörters. The parabolic Anderson model with heavy-tailed potential. In Surveys in stochastic processes, EMS Ser. Congr. Rep., pages 67–85. Eur. Math. Soc., Zürich, 2011.
  • [10] P. Mörters, M. Ortgiese, and N. Sidorova. Ageing in the parabolic Anderson model. Ann. Inst. Henri Poincaré Probab. Stat., 47(4):969–1000, 2011.
  • [11] M. Ortgiese and M. I. Roberts. Intermittency for branching random walk in Pareto environment. Ann. Probab., 44(3):2198–2263, 2016.
  • [12] M. Ortgiese and M. I. Roberts. Scaling limit and ageing for branching random walk in Pareto environment. Preprint arXiv:1602.08997, 2016.