Parabolic Anderson model with a finite number of moving catalysts

Abstract

We consider the parabolic Anderson model (PAM) which is given by the equation $\partial u/\partial t = κΔu + ξu$ with $u\colon\, \Z^d\times [0,\infty)\to \R$, where $κ\in [0,\infty)$ is the diffusion constant, $Δ$ is the discrete Laplacian, and $ξ\colon\,\Z^d\times [0,\infty)\to\R$ is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" $u$ under the influence of a "catalyst" $ξ$. In the present paper we focus on the case where $ξ$ is a system of $n$ independent simple random walks each with step rate $2dρ$ and starting from the origin. We study the \emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t.\ $ξ$ and show that these exponents, as a function of the diffusion constant $κ$ and the rate constant $ρ$, behave differently depending on the dimension $d$. In particular, we give a description of the intermittent behavior of the system in terms of the annealed Lyapunov exponents, depicting how the total mass of $u$ concentrates as $t\to\infty$. Our results are both a generalization and an extension of the work of Gärtner and Heydenreich 2006, where only the case $n=1$ was investigated.