Stationary parabolic Anderson model and intermittency

R. A. Carmona, S. A. Molchanov

Research output: Contribution to journalArticlepeer-review

29 Scopus citations

Abstract

This paper is devoted to the analysis of the large time behavior of the solutions of the Anderson parabolic problem: {Mathematical expression} 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.

Original languageEnglish (US)
Pages (from-to)433-453
Number of pages21
JournalProbability Theory and Related Fields
Volume102
Issue number4
DOIs
StatePublished - Dec 1 1995
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Mathematics Subject Classification: 60H25

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