Ageing in the parabolic Anderson model

Peter Mörters; Marcel Ortgiese; Nadia Sidorova

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 4, page 969-1000
  • ISSN: 0246-0203

Abstract

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The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model.

How to cite

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Mörters, Peter, Ortgiese, Marcel, and Sidorova, Nadia. "Ageing in the parabolic Anderson model." Annales de l'I.H.P. Probabilités et statistiques 47.4 (2011): 969-1000. <http://eudml.org/doc/240415>.

@article{Mörters2011,
abstract = {The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model.},
author = {Mörters, Peter, Ortgiese, Marcel, Sidorova, Nadia},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Anderson hamiltonian; parabolic problem; aging; random disorder; random medium; Heavy tail; extreme value theory; polynomial tail; Pareto distribution; point process; residual lifetime; scaling limit; functional limit theorem; Anderson Hamiltonian; heavy tail},
language = {eng},
number = {4},
pages = {969-1000},
publisher = {Gauthier-Villars},
title = {Ageing in the parabolic Anderson model},
url = {http://eudml.org/doc/240415},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Mörters, Peter
AU - Ortgiese, Marcel
AU - Sidorova, Nadia
TI - Ageing in the parabolic Anderson model
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 4
SP - 969
EP - 1000
AB - The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model.
LA - eng
KW - Anderson hamiltonian; parabolic problem; aging; random disorder; random medium; Heavy tail; extreme value theory; polynomial tail; Pareto distribution; point process; residual lifetime; scaling limit; functional limit theorem; Anderson Hamiltonian; heavy tail
UR - http://eudml.org/doc/240415
ER -

References

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  10. [10] N. Gantert, P. Mörters and V. Wachtel. Trap models with vanishing drift: Scaling limits and ageing regimes. ALEA Lat. Am. J. Probab. Math. Stat. To appear (2011). Available at arXiv:1003.1490. Zbl1276.60124MR2741195
  11. [11] J. Gärtner and S. A. Molchanov. Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm. Math. Phys. 132 (1990) 613–655. Zbl0711.60055MR1069840
  12. [12] W. König, H. Lacoin, P. Mörters and N. Sidorova. A two cities theorem for the parabolic Anderson model. Ann. Probab. 37 (2009) 347–392. Zbl1183.60024MR2489168
  13. [13] 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 (2008) 2450–2494. Zbl1204.60061MR2474543

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