  In this paper we study intermittency for the parabolic Anderson equation
$\partial u/\partial t=\kappa\Delta u+\gamma\xi u$ 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,
$\gamma\in(0,\infty)$ is the coupling constant, and
$\xi:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$ is a space--time random medium.
The solution of this equation describes the evolution of a ``reactant'' $u$
under the influence of a ``catalyst'' $\xi$. We focus on the case where $\xi$
is the voter model with opinions 0 and 1 that are updated according to a random
walk transition kernel, starting from either the Bernoulli measure $\nu_{\rho}$
or the equilibrium measure $\mu_{\rho}$, where $\rho\in(0,1)$ is the density of
1's. We consider the annealed Lyapunov exponents, that is, the exponential
growth rates of the successive moments of $u$. We show that if the random walk
transition kernel has zero mean and finite variance, then these exponents are
trivial for $1\leq d\leq4$, but display an interesting dependence on the
diffusion constant $\kappa$ for $d\geq 5$, with qualitatively different
behavior in different dimensions. In earlier work we considered the case where
$\xi$ is a field of independent simple random walks in a Poisson equilibrium,
respectively, a symmetric exclusion process in a Bernoulli equilibrium, which
are both reversible dynamics. In the present work a main obstacle is the
nonreversibility of the voter model dynamics, since this precludes the
application of spectral techniques. The duality with coalescing random walks is
key to our analysis, and leads to a representation formula for the Lyapunov
exponents that allows for the application of large deviation estimates.

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