Ageing in the parabolic Anderson model
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 969-1000.

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.

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.

DOI : https://doi.org/10.1214/10-AIHP394
Classification : 60K37,  82C44
Mots clés : 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
@article{AIHPB_2011__47_4_969_0,
     author = {M\"orters, Peter and Ortgiese, Marcel and Sidorova, Nadia},
     title = {Ageing in the parabolic Anderson model},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {969--1000},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {4},
     year = {2011},
     doi = {10.1214/10-AIHP394},
     zbl = {1268.82031},
     language = {en},
     url = {www.numdam.org/item/AIHPB_2011__47_4_969_0/}
}
Mörters, Peter; Ortgiese, Marcel; Sidorova, Nadia. Ageing in the parabolic Anderson model. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 969-1000. doi : 10.1214/10-AIHP394. http://www.numdam.org/item/AIHPB_2011__47_4_969_0/

[1] F. Aurzada and L. Döring. Intermittency and aging for the symbiotic branching model. Ann. Inst. Henri Poincaré Probab. Stat. To appear (2011). Available at arXiv:0905.1003. | EuDML 240811 | Numdam | MR 2814415 | Zbl 1222.60075

[2] G. Ben Arous and J. Černý. Bouchaud's model exhibits two different aging regimes in dimension one. Ann. Appl. Probab. 15 (2005) 1161-1192. | MR 2134101 | Zbl 1069.60092

[3] G. Ben Arous and J. Černý. The arcsine law as a universal aging scheme for trap models. Comm. Pure Appl. Math. 61 (2008) 289-329. | MR 2376843 | Zbl 1141.60075

[4] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1999. | MR 1700749 | Zbl 0172.21201

[5] J.-P. Bouchaud. Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France) 2 (1992) 1705-1713.

[6] A. Bovier and A. Faggionato. Spectral characterization of aging: The REM-like trap model. Ann. Appl. Probab. 15 (2005) 1997-2037. | MR 2152251 | Zbl 1086.60064

[7] J. Černý. The behaviour of aging functions in one-dimensional Bouchaud's trap model. Comm. Math. Phys. 261 (2006) 195-224. | MR 2193209 | Zbl 1107.81029

[8] A. Dembo and J.-D. Deuschel. Aging for interacting diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007) 461-480. | EuDML 77943 | Numdam | MR 2329512 | Zbl 1117.60088

[9] B. Fristedt and L. Gray. A Modern Approach to Probability Theory. Birkhäuser, Boston, MA, 1997. | MR 1422917 | Zbl 0869.60001

[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. | MR 2741195

[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. | MR 1069840 | Zbl 0711.60055

[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. | MR 2489168 | Zbl 1183.60024

[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. | MR 2474543 | Zbl 1204.60061