### Abstract

Original language | English |
---|---|

Pages (from-to) | 969-1000 |

Number of pages | 32 |

Journal | Annales de l'Institut Henri Poincaré, Probabilités et Statistiques |

Volume | 47 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2011 |

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### Cite this

*Annales de l'Institut Henri Poincaré, Probabilités et Statistiques*,

*47*(4), 969-1000. https://doi.org/10.1214/10-AIHP394

**Ageing in the parabolic Anderson model.** / Morters, Peter; Ortgiese, Marcel; Sidorova, Nadia.

Research output: Contribution to journal › Article

*Annales de l'Institut Henri Poincaré, Probabilités et Statistiques*, vol. 47, no. 4, pp. 969-1000. https://doi.org/10.1214/10-AIHP394

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TY - JOUR

T1 - Ageing in the parabolic Anderson model

AU - Morters, Peter

AU - Ortgiese, Marcel

AU - Sidorova, Nadia

PY - 2011

Y1 - 2011

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=80054111834&partnerID=8YFLogxK

UR - http://arxiv.org/abs/0910.5613

UR - http://dx.doi.org/10.1214/10-AIHP394

U2 - 10.1214/10-AIHP394

DO - 10.1214/10-AIHP394

M3 - Article

VL - 47

SP - 969

EP - 1000

JO - Annales de l'Institut Henri Poincaré: Probabilités et Statistiques

JF - Annales de l'Institut Henri Poincaré: Probabilités et Statistiques

SN - 0246-0203

IS - 4

ER -