Electronic Journal of Probability

Time Correlations for the Parabolic Anderson Model

Jürgen Gärtner and Adrian Schnitzler

Full-text: Open access

Abstract

We derive exact asymptotics of time correlation functions for the parabolic Anderson model with homogeneous initial condition and time-independent tails that decay more slowly than those of a double exponential distribution and have a finite cumulant generating function. We use these results to give precise asymptotics for statistical moments of positive order. Furthermore, we show what the potential peaks that contribute to the intermittency picture look like and how they are distributed in space. We also investigate for how long intermittency peaks remain relevant in terms of ageing properties of the model.

Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 56, 1519-1548.

Dates
Accepted: 20 August 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464820225

Digital Object Identifier
doi:10.1214/EJP.v16-917

Mathematical Reviews number (MathSciNet)
MR2827469

Zentralblatt MATH identifier
1245.60100

Subjects
Primary: 60K37: Processes in random environments 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Secondary: 60H25: Random operators and equations [See also 47B80]

Keywords
Parabolic Anderson model Anderson Hamiltonian random potential time correlations annealed asymptotics intermittency ageing

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Gärtner, Jürgen; Schnitzler, Adrian. Time Correlations for the Parabolic Anderson Model. Electron. J. Probab. 16 (2011), paper no. 56, 1519--1548. doi:10.1214/EJP.v16-917. https://projecteuclid.org/euclid.ejp/1464820225


Export citation

References

  • F. Aurzada, L. Döring. Intermittency and Aging for the Symbiotic Branching Model. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), 376-394.
  • G. Ben Arous. Aging and spin-glass dynamics. Proceedings of the International Congress of Mathematicians Vol. III, (2002), 3-14. Higher Ed. Press, Beijing.
  • 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, AMS/CRM. 42 (2007), 1-40.
  • N.-H. Bingham, C.H. Goldie and J.L. Teugels. Regular Variation, Cambridge University Press (1987).
  • C. Carmona, S.A. Molchanov. Parabolic Anderson Problem and intermittency. AMS Memoir 518 (1994) American Mathematical Society.
  • A. Dembo, J.-D. Deuschel. Aging for interacting dif and only ifusion processes, Ann. Inst. H. Poincaré Probab. Statist. 43 (2007), 461-480.
  • P. Feigin, E. Yashchin. On a strong Tauberian result, Probab. Theory Related Fields 65 (1983), 35-48.
  • J. Gärtner, F. den Hollander. Correlation structure of intermittency in the parabolic Anderson model. Probab. Theory Related Fields 114 (1999), 1-54.
  • J. Gärtner, W. König. The parabolic Anderson model. in: J.-D. Deuschel and A. Greven (Eds.), Interacting Stochastic Systems, Springer (2005), 153-179.
  • J. Gärtner, W. König and S.A. Molchanov, Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35 (2007), 439-499.
  • J. Gärtner, S.A. Molchanov. Parabolic problems for the Anderson model: I. Intermittency and related topics. Commun. Math. Phys. 132 (1990), 613-655.
  • J. Gärtner, 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.
  • J. Gärtner, S.A. Molchanov. Moment asymptotics and Lifshitz tails for the parabolic Anderson model. In Canadian Math. Soc. Conference Proceedings 26 (L. G. Gorostiza and B. G. Ivanoff, Eds.), Amer. Math. Soc. (2000), 141-157.
  • R. van der Hofstad, W. König and P. Mörters. The universality classes in the parabolic Anderson model. Comm. Math. Phys. 267 (2006), 307-353.
  • W. Kirsch, An Invitation to Random Schrödinger operators. arXiv:0709.3707v1 (2007).
  • W. König, H. Lacoin, P. Mörters N. Sidorova. A two cities theorem for the parabolic Anderson model. Ann. Probab. 37 (2009), 347-392.
  • S.A. Molchanov. Lectures on Random Media, Lecture Notes in Math. 1581, Springer (1994), 242-411.
  • P. Mörters, M. Ortgiese and N. Sidorova, Ageing in the parabolic Anderson model, arXiv:0910.5613v1 (2009).