Open Access
November 2006 Intermittency in a catalytic random medium
J. Gärtner, F. den Hollander
Ann. Probab. 34(6): 2219-2287 (November 2006). DOI: 10.1214/009117906000000467


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→∞.


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J. Gärtner. F. den Hollander. "Intermittency in a catalytic random medium." Ann. Probab. 34 (6) 2219 - 2287, November 2006.


Published: November 2006
First available in Project Euclid: 13 February 2007

zbMATH: 1117.60065
MathSciNet: MR2294981
Digital Object Identifier: 10.1214/009117906000000467

Primary: 60H25 , 82C44
Secondary: 35B40 , 60F10

Keywords: catalytic behavior , catalytic random medium , Intermittency , large deviations , Parabolic Anderson model

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 6 • November 2006
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