A Scaling Limit Theorem for the Parabolic Anderson Model with Exponential Potential

Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 11)

Abstract

The parabolic Anderson problem is the Cauchy problem for the heat equation with random potential and localized initial condition. In this paper, we consider potentials which are constant in time and independent exponentially distributed in space. We study the growth rate of the total mass of the solution in terms of weak and almost sure limit theorems, and the spatial spread of the mass in terms of a scaling limit theorem. The latter result shows that in this case, just like in the case of heavy tailed potentials, the mass gets trapped in a single relevant island with high probability.

Keywords

Variational Problem Moment Generate Function Continuous Time Random Walk Exponential Potential Brownian Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Biskup, M., König, W.: Eigenvalue order statistics for random Schrödinger operators with doubly exponential tails. In preparation (2010)Google Scholar
  2. 2.
    Caravenna, F.,  Carmona, P., Pétrélis, N.: The discrete-time parabolic Anderson model with heavy-tailed potential. Preprint arXiv:1012.4653v1[math.PR] (2010)Google Scholar
  3. 3.
    Gärtner, J., König, W.,  Molchanov, S.: Geometric characterisation of intermittency in the parabolic Anderson model. Ann. Probab. 35, 439–499 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Gärtner, J., Molchanov, S.: Parabolic problems for the Anderson model. I. Intermittency and related topics. Commun. Math. Phys. 132, 613–655 (1990)MATHCrossRefGoogle Scholar
  5. 5.
    Gärtner, J., Molchanov, S.: Parabolic problems for the Anderson model. II. Second-order asymptotics and structure of high peaks. Probab. Theory Relat. Fields 111, 17–55 (1998)MATHCrossRefGoogle Scholar
  6. 6.
    van der Hofstad, R., König, W., Mörters, P.: The universality classes in the parabolic Anderson model. Commun. Math. Phys. 267, 307–353 (2006)MATHCrossRefGoogle Scholar
  7. 7.
    van der Hofstad, R., Mörters, P., Sidorova, N.: Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials. Ann. Appl. Probab. 18, 2450–2494 (2008)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    König, W., Lacoin, H., Mörters, P., Sidorova, N.: A two cities theorem for the parabolic Anderson model. Ann. Probab. 37, 347–392 (2009)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Lacoin, H.: Calcul d’asymptotique et localization p.s. pour le modèle parabolique d’Anderson. Mémoire de Magistère, ENS, Paris (2007)Google Scholar
  10. 10.
    Mörters, P.: The parabolic Anderson model with heavy-tailed potential. In: Surveys in Stochastic Processes. Proceedings of the 33rd SPA Conference in Berlin, 2009. Edited by J. Blath, P. Imkeller, and S. Roelly. pp. 67–85. EMS Series of Congress Reports (2011)Google Scholar
  11. 11.
    Mörters, P., Ortgiese, M., Sidorova, N.: Ageing in the parabolic Anderson model. Annales de l’Institut Henri Poincaré: Probab. et Stat. 47, 969–1000 (2011)CrossRefGoogle Scholar
  12. 12.
    Mörters, P.: The parabolic Anderson model with heavy-tailed potential. To appear in: ‘Surveys in Stochastic Processes’. J. Blath et al. (eds.) EMS Conference Reports (2011)Google Scholar
  13. 13.
    Resnick, S.I.: Extreme values, regular variation, and point processes. Springer Series in OR and Financial Engineering. Springer, New York (1987)MATHGoogle Scholar
  14. 14.
    Sznitman, A.-S.: Crossing velocities and random lattice animals. Ann. Probab. 23, 1006–1023 (1995)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Sznitman, A.-S.: Fluctuations of principal eigenvalues and random scales. Commun. Math. Phys. 189, 337–363 (1997)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Sznitman, A.-S.: Brownian motion, obstacles and random media. Springer, New York (1998)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.CEREMADE, Université Paris DauphineParisFrance
  2. 2.University of BathBathUK

Personalised recommendations