Journal of Statistical Physics

, Volume 129, Issue 1, pp 151–169 | Cite as

Anderson Parabolic Model for a Quasi-Stationary Medium

  • C. BoldrighiniEmail author
  • S. Molchanov
  • A. Pellegrinotti


We study the Anderson Parabolic Model for a random medium which is a product of an i.i.d. space-like random field and a white noise. The model has long range space-time correlations and is intermediate between the stationary case and the “turbulent” one, which were studied in previous works. Under some natural assumptions on the distribution of the space potential, we prove existence and uniqueness, and derive the long time asymptotics for the annealed moments, and the “semi-annealed” ones, for which expectation is taken only w.r.t. the white noise. A conjecture for the fully quenched case is discussed on a simplified model.


Random Walk White Noise Weibull Distribution Stat Phys Random Medium 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Gärtner, J., Molchanov, S.A.: Parabolic problems for the Anderson model. I. Intermittency and related topics. Commun. Math. Phys. 132(3), 613–655 (1990) zbMATHCrossRefADSGoogle Scholar
  2. 2.
    Gärtner, J., Molchanov, S.A.: Parabolic problems for the Anderson model. II. Second-order asymptotics and structure of high peaks. Probab. Theory Related Fields 111(1), 17–55 (1998) zbMATHCrossRefGoogle Scholar
  3. 3.
    Zeldovich, Ya., Molchanov, S., Ruzmaikin, A., Sokolov, D.: Intermittency, Diffusion and Generation in a Non-Stationary Random Medium. Russian Reviews in Mathematical Physics. Cambridge University Press, Cambridge (1989) Google Scholar
  4. 4.
    Carmona, R., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Mem. Am. Math. Soc. 108, 518 (1994) Google Scholar
  5. 5.
    Molchanov, S.: Lectures on Random Media. In: Bernard, P. (ed.) Lectures in Probability Theory, Ecole d’ Eté de Probabilités de Saint Flour XXII, 1992. Lecture Notes in Mathematics, vol. 1581, pp. 242–411. Springer, Berlin (1994) Google Scholar
  6. 6.
    Boldrighini, C., Minlos, R.A., Pellegrinotti, A.: Random walks in quenched i.i.d. space–time random environment are always a.s. diffusive. Probab. Theory Relat. Fields 129(1), 133–156 (2004) zbMATHCrossRefGoogle Scholar
  7. 7.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and Applications (No. 27). Cambridge University Press, Cambridge (1989) zbMATHGoogle Scholar
  8. 8.
    Ben Arous, G., Bogachev, L., Molchanov, S.A.: Limit theorems for sums of random exponentials. Probab. Theory Relat. Fields 132, 579–612 (2005) zbMATHCrossRefGoogle Scholar
  9. 9.
    Ross, S.: Introduction to Probability Models. Academic Press, Orlando (1985) zbMATHGoogle Scholar
  10. 10.
    Gihman, I.I., Skorohod, A.V.: Stochastic Differential Equations. Springer, Berlin (1972) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • C. Boldrighini
    • 1
    Email author
  • S. Molchanov
    • 2
  • A. Pellegrinotti
    • 3
  1. 1.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomeItaly
  2. 2.Department of Mathematics and StatisticsUniversity of North Carolina at CharlotteCharlotteUSA
  3. 3.Dipartimento di MatematicaUniversità di Roma TreRomeItaly

Personalised recommendations