The Annals of Probability

Intermittency in a catalytic random medium

J. Gärtner and F. den Hollander

Full-text: Open access

Abstract

In this paper, we study intermittency for the parabolic Anderson equation ∂u/∂t=κΔu+ξu, where u:ℤd×[0, ∞)→ℝ, κ is the diffusion constant, Δ is the discrete Laplacian and ξ:ℤd×[0, ∞)→ℝ is a space-time random medium. We focus on the case where ξ is γ times the random medium that is obtained by running independent simple random walks with diffusion constant ρ starting from a Poisson random field with intensity ν. Throughout the paper, we assume that κ, γ, ρ, ν∈(0, ∞). The solution of the equation describes the evolution of a “reactant” u under the influence of a “catalyst” ξ.

We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of u, and show that they display an interesting dependence on the dimension d and on the parameters κ, γ, ρ, ν, with qualitatively different intermittency behavior in d=1, 2, in d=3 and in d≥4. Special attention is given to the asymptotics of these Lyapunov exponents for κ↓0 and κ→∞.

Article information

Source
Ann. Probab. Volume 34, Number 6 (2006), 2219-2287.

Dates
First available in Project Euclid: 13 February 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1171377442

Digital Object Identifier
doi:10.1214/009117906000000467

Mathematical Reviews number (MathSciNet)
MR2294981

Zentralblatt MATH identifier
1117.60065

Subjects
Primary: 60H25: Random operators and equations [See also 47B80] 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Secondary: 60F10: Large deviations 35B40: Asymptotic behavior of solutions

Keywords
Parabolic Anderson model catalytic random medium catalytic behavior intermittency large deviations

Citation

Gärtner, J.; den Hollander, F. Intermittency in a catalytic random medium. Ann. Probab. 34 (2006), no. 6, 2219--2287. doi:10.1214/009117906000000467. http://projecteuclid.org/euclid.aop/1171377442.


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