Intermittency in a catalytic random medium

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→∞.