  We continue our study of the parabolic Anderson equation $\partial u/\partial
t = \kappa\Delta u + \gamma\xi u$ for the space-time field $u\colon\,\Z^d\times
[0,\infty)\to\R$, where $\kappa \in [0,\infty)$ is the diffusion constant,
$\Delta$ is the discrete Laplacian, $\gamma\in (0,\infty)$ is the coupling
constant, and $\xi\colon\,\Z^d\times [0,\infty)\to\R$ 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" $\xi$,
both living on $\Z^d$. In earlier work we considered three choices for $\xi$:
independent simple random walks, the symmetric exclusion process, and the
symmetric voter model, all in equilibrium at a given density. We analyzed the
\emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the
successive moments of $u$ w.r.t.\ $\xi$, and showed that these exponents
display an interesting dependence on the diffusion constant $\kappa$, with
qualitatively different behavior in different dimensions $d$. In the present
paper we focus on the \emph{quenched} Lyapunov exponent, i.e., the exponential
growth rate of $u$ conditional on $\xi$. We first prove existence and derive
some qualitative properties of the quenched Lyapunov exponent for a general
$\xi$ that is stationary and ergodic w.r.t.\ translations in $\Z^d$ and
satisfies certain noisiness conditions. After that we focus on the three
particular choices for $\xi$ mentioned above and derive some more detailed
properties. We close by formulating a number of open problems.
