Probability Theory and Related Fields

, Volume 102, Issue 4, pp 433–453

Stationary parabolic Anderson model and intermittency

  • R. A. Carmona
  • S. A. Molchanov
Article

Summary

This paper is devoted to the analysis of the large time behavior of the solutions of the Anderson parabolic problem:
$$\frac{{\partial u}}{{\partial t}} = \kappa \Delta u\xi (x)u$$
when the potential ξ(x) is a homogeneous ergodic random field on ℝd. Our goal is to prove the asymptotic spatial intermittency of the solution and for this reason, we analyze the large time properties of all the moments of the positive solutions. This provides an extension to the continuous space ℝd of the work done originally by Gärtner and Molchanov in the case of the lattice ℤd. In the process of our moment analysis, we show that it is possible to exhibit new asymptotic regimes by considering a special class of generalized Gaussian fields, interpolating continuously between the exponent 2 which is found in the case of bona fide continuous Gaussian fields ξ(x) and the exponent 3/2 appearing in the case of a one dimensional white noise. Finally, we also determine the precise almost sure large time asymptotics of the positive solutions.

Mathematics Subject Classification

60H25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler, R.J.: An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. IMS Lect. Notes # 12. Hayward (1990)Google Scholar
  2. 2.
    Ahn, H.S., Carmona, R., Molchanov, S.A.: Parabolic equations with a Lévy random potential. Proc. Charlotte Conf. on Stochast. Part. Diff. Equations, 1991 (to appear)Google Scholar
  3. 3.
    Aizenman, M., Molchanov, S.A.: Localization at large disorder and extreme energies: an elementary derivation (preprint, 1993)Google Scholar
  4. 4.
    Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Boston: Birkhaüser 1990Google Scholar
  5. 5.
    Carmona, R., Molchanov, S.A.: Intermittency and phase transitions for some particle systems in random media. Proc. Katata Symp, June 1990 (to appear)Google Scholar
  6. 6.
    Carmona, R., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Memoirs Amer. Math. Soc.108 #514 (1994)Google Scholar
  7. 7.
    Carmona, R., Grishin, S., Molchanov, S.A.: Asymptotic for the boundary parabolic Anderson problem in a half space (preprint, 1994).Google Scholar
  8. 8.
    Donsker, M.D., Varadhan, S.R.S.: Asymtotics for the Wiener saussage. Commun. Pure and Appl. Math28, 525–565 (1975)Google Scholar
  9. 9.
    Donsker, M., Varadhan, S.R.S.: Asymptotics for the Polaron. Commun. Pure Appl. Math.36, 183–212 (1983)Google Scholar
  10. 10.
    Fernique, X.: Regularité des trajectoires des fonctions aléatoires gaussiennes. (Lect. Notes Math., Vol.480, 1–96 (1975)Google Scholar
  11. 11.
    Figotin, A., Pastur, L.A.: Spectral properties of disordered systems in the one body approximation. New York: Springer 1993Google Scholar
  12. 12.
    Gärtner, J., Molchanov, S.A.: Parabolic problems for the Anderson model. Comm. Math. Phys.132, 613–655 (1990)Google Scholar
  13. 13.
    Guelfand, I.M., Vilenkin, G.: Generalized Functions, Vol. 4. New York: Academic Press 1964Google Scholar
  14. 14.
    Klein A., Landau L.J.: Construction of a unique self-adjoint generator for a symmetric local semigroup. J. Funct. Anal.44, 121–137 (1981)Google Scholar
  15. 15.
    Lifschitz, I.M., Gredeskul, S.A., Pastur, L.A.: Introduction to the theory of disordered systems. New York: Wiley 1988Google Scholar
  16. 16.
    Mansmann, U.: Strong coupling limit for a certain class of polaron models. Probab. Theory Relat. Fields90, 427–446 (1991)Google Scholar
  17. 17.
    Molchanov, S.A.: Ideas in the theory of random media. Acta Applicandae Math.22, 139–282 (1990)Google Scholar
  18. 18.
    Reed, M., Simon, B.: Methods of modern mathematical physics II: Fourier analysis, self-adjointness. New York: Academic Press 1975Google Scholar
  19. 19.
    Sznitman, A.S.: Brownian asymptotics in a Poissonian environment. Probab. Theory Relat. Fields95, 155–174 (1993)Google Scholar
  20. 20.
    Weber, M.: Sur le comportement Asymptotique des Processus Gaussiens Stationaires. in Aspects Statistiques et Aspects Physiques dse Processus Gaussiens. Paris: CNRS pp. 563–567 1981Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • R. A. Carmona
    • 1
  • S. A. Molchanov
    • 2
  1. 1.Department of MathematicsUniversity of California at IrvineIrvineUSA
  2. 2.Department of MathematicsUniversity of North Carolina at CharlotteCharlotteUSA

Personalised recommendations