The Annals of Applied Probability

Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials

Remco van der Hofstad, Peter Mörters, and Nadia Sidorova

Full-text: Open access


We study the parabolic Anderson problem, that is, the heat equation tuu+ξu on (0, ∞)×ℤd with independent identically distributed random potential {ξ(z) : z∈ℤd} and localized initial condition u(0, x)=10(x). Our interest is in the long-term behavior of the random total mass U(t)=∑zu(t, z) of the unique nonnegative solution in the case that the distribution of ξ(0) is heavy tailed. For this, we study two paradigm cases of distributions with infinite moment generating functions: the case of polynomial or Pareto tails, and the case of stretched exponential or Weibull tails. In both cases we find asymptotic expansions for the logarithm of the total mass up to the first random term, which we describe in terms of weak limit theorems. In the case of polynomial tails, already the leading term in the expansion is random. For stretched exponential tails, we observe random fluctuations in the almost sure asymptotics of the second term of the expansion, but in the weak sense the fourth term is the first random term of the expansion. The main tool in our proofs is extreme value theory.

Article information

Ann. Appl. Probab., Volume 18, Number 6 (2008), 2450-2494.

First available in Project Euclid: 26 November 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H25: Random operators and equations [See also 47B80] 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Secondary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60G70: Extreme value theory; extremal processes

Anderson Hamiltonian parabolic Anderson problem long term behavior intermittency localization random environment random potential partial differential equations with random coefficients heavy tails extreme value theory Pareto distribution Weibull distribution weak limit theorem law of the iterated logarithm


van der Hofstad, Remco; Mörters, Peter; Sidorova, Nadia. Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials. Ann. Appl. Probab. 18 (2008), no. 6, 2450--2494. doi:10.1214/08-AAP526.

Export citation


  • [1] Antal, P. (1995). Enlargement of obstacles for the simple random walk. Ann. Probab. 23 1061–1101.
  • [2] Ben Arous, G., Molchanov, S. and Ramírez, A. F. (2005). Transition from the annealed to the quenched asymptotics for a random walk on random obstacles. Ann. Probab. 33 2149–2187.
  • [3] Biskup, M. and König, W. (2001). Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29 636–682.
  • [4] Carmona, R. A. and Molchanov, S. A. (1994). Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 viii+125.
  • [5] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Applications of Mathematics (New York) 33. Springer, Berlin.
  • [6] Gärtner, J. and König, W. (2005). The parabolic Anderson model. In Interacting Stochastic Systems 153–179. Springer, Berlin.
  • [7] Gärtner, J., König, W. and Molchanov, S. (2007). Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35 439–499.
  • [8] Gärtner, J. and Molchanov, S. A. (1990). Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm. Math. Phys. 132 613–655.
  • [9] Gärtner, J. and Molchanov, S. A. (1998). Parabolic problems for the Anderson model. II. Second-order asymptotics and structure of high peaks. Probab. Theory Related Fields 111 17–55.
  • [10] van der Hofstad, R., König, W. and Mörters, P. (2006). The universality classes in the parabolic Anderson model. Comm. Math. Phys. 267 307–353.
  • [11] Lacoin, H. (2007). Calcul d’asymptotique et localization p.s. pour le modèle parabolique d’Anderson. Mémoire de Magistère, ENS, Paris.
  • [12] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  • [13] Zel’dovich, Y. B., Molchanov, S. A., Ruzmaĭkin, A. A. and Sokolov, D. D. (1987). Intermittency in random media. Uspekhi Fiz. Nauk 152 3–32.