Intermittency in a catalytic random medium

Abstract

In this paper, we study intermittency for the parabolic Anderson equation $\partial u/\partial t=κΔu+ξu$, where $u:\mathbb{Z}^d\times [0,\infty)\to\mathbb{R}$, $κ$ is the diffusion constant, $Δ$ is the discrete Laplacian and $ξ:\mathbb{Z}^d\times[0,\infty)\to\mathbb {R}$ 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 $κ,γ,ρ,ν\in (0,\infty)$. 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 $d\geq4$. Special attention is given to the asymptotics of these Lyapunov exponents for $κ\downarrow0$ and $κ\to\infty$.