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We continue our study of the parabolic Anderson equation $\partial u/\partialt = \kappa\Delta u + \gamma\xi u$ 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 couplingconstant, and $\xi\colon\,\Z^d\times [0,\infty)\to\R$ is a space-time randomenvironment that drives the equation. The solution of this equation describesthe 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 thesymmetric voter model, all in equilibrium at a given density. We analyzed the\emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of thesuccessive moments of $u$ w.r.t.\ $\xi$, and showed that these exponentsdisplay an interesting dependence on the diffusion constant $\kappa$, withqualitatively different behavior in different dimensions $d$. In the presentpaper we focus on the \emph{quenched} Lyapunov exponent, i.e., the exponentialgrowth rate of $u$ conditional on $\xi$. We first prove existence and derivesome qualitative properties of the quenched Lyapunov exponent for a general$\xi$ that is stationary and ergodic w.r.t.\ translations in $\Z^d$ andsatisfies certain noisiness conditions. After that we focus on the threeparticular choices for $\xi$ mentioned above and derive some more detailedproperties. We close by formulating a number of open problems.

Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment