m at h . PR ] 1 4 Ju n 20 04 Intermittency in a catalytic random medium
نویسندگان
چکیده
In this paper we study intermittency for the parabolic Anderson equation ∂u/∂t = κ∆u + ξu, where u : Z d × [0, ∞) → R, κ is the diffusion constant, ∆ is the discrete Laplacian, and ξ : Z d × [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. 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, i.e., 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 and 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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