Parabolic Anderson model with a finite number of moving catalysts

Abstract

We consider the parabolic Anderson model (PAM) which is given by the equation $\partial u/\partial t=\kappa \Delta u+\xi u$ with $u:{\mathbb{Z}}^d\times [0,\infty )\to \mathbb{R}$, where $\kappa \in [0,\infty )$ is the diffusion constant, $\Delta$ is the discrete Laplacian, and $\xi :{\mathbb{Z}}^d\times [0,\infty )\to \mathbb{R}$ 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" $\xi$. In the present paper we focus on the case where $\xi$ is a system of $n$ independent simple random walks each with step rate $2d\rho$ 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.\ $\xi$ and show that these exponents, as a function of the diffusion constant $\kappa$ and the rate constant $\rho$, 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.

arXiv commentIn honour of Jürgen Gärtner on the occasion of his 60th birthday, 25 pages. Updated version following the referee's comments