Parabolic Anderson model with a finite number of moving catalysts

We consider the parabolic Anderson model (PAM) which is given by the equation $\mathrm{\partial }u\mathrm{/}\mathrm{\partial }t=\kappa \mathrm{\Delta }u+\xi u$ with $u\colon, \Z^d\times [0,\infty)\to \R$,where$\kappa \in [0,\infty)$isthediffusionconstant,$\Delta$isthediscreteLaplacian,and$\xi\colon,\Z^d\times [0,\infty)\to\R$ 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” $\xi$. 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 $t\to \mathrm{\infty }$. Our results are both a generalization and an extension of the work of G"artner and Heydenreich 2006, where only the case $n=1$ was investigated.