Time Correlations for the Parabolic Anderson Model

Jürgen Gärtner (Technische Universität Berlin)
Adrian Schnitzler (Technische Universität Berlin)


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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1519-1548

Publication Date: August 20, 2011

DOI: 10.1214/EJP.v16-917


  1. F. Aurzada, L. Döring. Intermittency and Aging for the Symbiotic Branching Model. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), 376-394.
  2. G. Ben Arous. Aging and spin-glass dynamics. Proceedings of the International Congress of Mathematicians Vol. III , (2002), 3-14. Higher Ed. Press, Beijing. MR1957514
  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, AMS/CRM. 42 (2007), 1-40. MR2352279
  4. N.-H. Bingham, C.H. Goldie and J.L. Teugels. Regular Variation, Cambridge University Press (1987). MR0898871
  5. C. Carmona, S.A. Molchanov. Parabolic Anderson Problem and intermittency. AMS Memoir 518 (1994) American Mathematical Society. MR1185878
  6. A. Dembo, J.-D. Deuschel. Aging for interacting dif and only ifusion processes, Ann. Inst. H. Poincaré Probab. Statist. 43 (2007), 461-480. MR2329512
  7. P. Feigin, E. Yashchin. On a strong Tauberian result, Probab. Theory Related Fields 65 (1983), 35-48. MR0717931
  8. J. Gärtner, F. den Hollander. Correlation structure of intermittency in the parabolic Anderson model. Probab. Theory Related Fields 114 (1999), 1-54. MR1697138
  9. J. Gärtner, W. König. The parabolic Anderson model. in: J.-D. Deuschel and A. Greven (Eds.), Interacting Stochastic Systems, Springer (2005), 153-179. MR2118574
  10. J. Gärtner, W. König and S.A. Molchanov, Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35 (2007), 439-499. MR2308585
  11. J. Gärtner, S.A. Molchanov. Parabolic problems for the Anderson model: I. Intermittency and related topics. Commun. Math. Phys. 132 (1990), 613-655. MR1069840
  12. 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. MR1626766
  13. 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. MR1765008
  14. R. van der Hofstad, W. König and P. Mörters. The universality classes in the parabolic Anderson model. Comm. Math. Phys. 267 (2006), 307-353. MR2249772
  15. W. Kirsch, An Invitation to Random Schrödinger operators. arXiv:0709.3707v1 (2007). MR2509110
  16. W. König, H. Lacoin, P. Mörters N. Sidorova. A two cities theorem for the parabolic Anderson model. Ann. Probab. 37 (2009), 347-392. MR2489168
  17. S.A. Molchanov. Lectures on Random Media, Lecture Notes in Math. 1581, Springer (1994), 242-411. MR1307415
  18. P. Mörters, M. Ortgiese and N. Sidorova, Ageing in the parabolic Anderson model, arXiv:0910.5613v1 (2009).

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.