Intermittency in a catalytic random medium
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J. Gärtner [2] ; F. den Hollander [1]
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[1] Leiden University
Leiden University
Países Bajos
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[2] Techniche Universität Berlin
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 34, Nº. 6, 2006, págs. 2219-2287
- Idioma: inglés
- DOI: 10.1214/009117906000000467
- Texto completo no disponible (Saber más ...)
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Resumen
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→∞.
