Probability Theory and Related Fields

, Volume 102, Issue 4, pp 433–453

Stationary parabolic Anderson model and intermittency

Authors

  • R. A. Carmona
    • Department of MathematicsUniversity of California at Irvine
  • S. A. Molchanov
    • Department of MathematicsUniversity of North Carolina at Charlotte
Article

DOI: 10.1007/BF01198845

Cite this article as:
Carmona, R.A. & Molchanov, S.A. Probab. Th. Rel. Fields (1995) 102: 433. doi:10.1007/BF01198845

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

Copyright information

© Springer-Verlag 1995