Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Ageing in the parabolic Anderson model

Peter Mörters, Marcel Ortgiese, and Nadia Sidorova

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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.

Résumé

Le modèle parabolique d’Anderson est le problème de Cauchy pour l’équation de la chaleur avec un potentiel aléatoire. Nous considérons ce modèle en temps continu et espace discret. Nous nous intéressons à des potentiels constants dans le temps, indépendants et identiquement distribués avec queues polynomiales à l’infini. Nous étudions les dynamiques temporelles à temps long de ce système. Notre résultat principal est que les périodes durant lesquelles le profil des solutions reste presque constant, croissent linéairement au cours du temps, un phénomène connu sous le nom de vieillissement. Nous décrivons ce phénomène au sens faible, en étudiant la probabilité asymptotique d’un changement dans un intervalle de temps donné, ainsi qu’au sens fort, en identifiant l’enveloppe supérieure presque sûre pour le processus du temps restant jusqu’au prochain changement du profil. Finalement nous démontrons des théorèmes de limite d’échelle fonctionnelle pour le profil et le taux de croissance de la solution du modèle parabolique d’Anderson.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 4 (2011), 969-1000.

Dates
First available in Project Euclid: 6 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1317906497

Digital Object Identifier
doi:10.1214/10-AIHP394

Mathematical Reviews number (MathSciNet)
MR2884220

Zentralblatt MATH identifier
1268.82031

Subjects
Primary: 60K37: Processes in random environments
Secondary: 82C44: Dynamics of disordered systems (random Ising systems, etc.)

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

Citation

Mörters, Peter; Ortgiese, Marcel; Sidorova, Nadia. Ageing in the parabolic Anderson model. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 4, 969--1000. doi:10.1214/10-AIHP394. https://projecteuclid.org/euclid.aihp/1317906497


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