Stable limit laws for the parabolic Anderson model between quenched and annealed behaviour

Gärtner, Jürgen; Schnitzler, Adrian

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 194-206.

Nous considérons la solution du modèle parabolique d’Anderson avec condition initiale homogène sur de grandes boîtes dépendantes du temps. Nous dérivons des théorèmes limites stables, pour toutes les valeurs possibles des paramètres d’échelle, pour la somme de la solution changée d’échelle en fonction du taux de croissance des boîtes. De plus, nous donnons des conditions suffisantes pour une loi des grands nombres.

We consider the solution to the parabolic Anderson model with homogeneous initial condition in large time-dependent boxes. We derive stable limit theorems, ranging over all possible scaling parameters, for the rescaled sum over the solution depending on the growth rate of the boxes. Furthermore, we give sufficient conditions for a strong law of large numbers.

[1] G. Ben Arous, L. V. Bogachev and S. A. Molchanov. Limit theorems for sums of random exponentials. Probab. Theory Related Fields 132 (2005) 579–612. | DOI | MR | Zbl

[2] G. Ben Arous, S. Molchanov and A. Ramirez. Transition from the annealed to the quenched asymptotics for a random walk on random obstacles. Ann. Probab. 33 (2005) 2149–2187. | DOI | MR | Zbl

[3] G. Ben Arous, S. Molchanov and A. Ramirez. Transition asymptotics for reaction–diffusion in random media. In Probability and Mathematical Physics: A Volume in Honor of Stanislav Molchanov. CRM Proc. Lecture Notes 42 1–40. Amer. Math. Soc., Providence, RI, 2007. | MR | Zbl

[4] M. Biskup and W. König. Long-time tails for the parabolic Anderson model with bounded potential. Ann. Probab. 29 (2001) 636–682. | DOI | MR | Zbl

[5] L. Bogachev. Limit laws for norms of IID samples with Weibull tails. J. Theoret. Probab. 19 (2006) 849–873. | DOI | MR | Zbl

[6] A. Bovier, I. Kurkova and M. Löwe. Fluctuations of the free energy in the REM and the $p$-spin SK models. Ann. Probab. 30 (2002) 605–651. | DOI | MR | Zbl

[7] M. Cranston and S. A. Molchanov. Quenched to annealed transition in the parabolic Anderson problem. Probab. Theory Related Fields 138 (2007) 177–193. | DOI | MR | Zbl

[8] J. Gärtner and W. König. The parabolic Anderson model. In Interacting Stochastic Systems 153–179. J.-D. Deuschel and A. Greven (Eds). Springer, Berlin, 2005. | MR | Zbl

[9] 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 | Zbl

[10] J. Gärtner and S. A. Molchanov. Parabolic problems for the Anderson model. II. Second-order asymptotics and structure of high peaks. Probab. Theory Related Fields 111 (1998) 17–55. | MR | Zbl

[11] J. Gärtner and A. Schnitzler. Time correlations for the parabolic Anderson model. Electron. J. Probab. 16 (2011) 1519–1548. | DOI | MR | Zbl

[12] R. Van Der Hofstad, W. König and P. Mörters. The universality classes in the parabolic Anderson model. Comm. Math. Phys. 267 (2006) 307–353. | DOI | MR | Zbl

[13] A. Janssen. Limit laws for power sums and norms of i.i.d. samples. Probab. Theory Related Fields 146 (2010) 515–533. | DOI | MR | Zbl

[14] H. Lacoin and P. Mörters. A scaling limit theorem for the parabolic Anderson model with exponential potential. In Probability in Complex Physical Systems. In Honour of J. Gärtner and E. Bolthausen 247–271. Springer Proceedings in Mathematics 11. Springer, Berlin, 2012. DOI:10.1007/978-3-642-23811-6_10. | MR | Zbl

[15] S. A. Molchanov. Lectures on random media. Lecture Notes in Math. 1581 242–411. Springer, Berlin, 1994. | MR | Zbl

[16] V. V. Petrov. Sums of Independent Random Variables. Springer, New York, 1975. | MR | Zbl

[17] N. Sidorova and A. Twarowski. Localisation and ageing in the parabolic Anderson model with Weibull potential. Ann. Probab. To appear. Available at arXiv:1204.1233v2, 2012. | MR | Zbl