  In this paper, we study intermittency for the parabolic Anderson equation
$\partial u/\partial t=\kappa\Delta u+\xi u$, where $u:\mathbb{Z}^d\times
[0,\infty)\to\mathbb{R}$, $\kappa$ 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 medium. We focus on the case where $\xi$ is $\gamma$ times
the random medium that is obtained by running independent simple random walks
with diffusion constant $\rho$ starting from a Poisson random field with
intensity $\nu$. Throughout the paper, we assume that
$\kappa,\gamma,\rho,\nu\in (0,\infty)$. The solution of the equation describes
the evolution of a ``reactant'' $u$ under the influence of a ``catalyst''
$\xi$. 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
$\kappa,\gamma,\rho,\nu$, 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 $\kappa\downarrow0$ and $\kappa
\to\infty$.
