  We consider the solution $u\colon [0,\infty) \times\mathbb{Z}^d\rightarrow
[0,\infty) $ to the parabolic Anderson model, where the potential is given by
$(t,x)\mapsto\gamma\delta_{Y_t}(x)$ with $Y$ a simple symmetric random walk on
$\mathbb{Z}^d$. Depending on the parameter $\gamma\in[-\infty,\infty)$, the
potential is interpreted as a randomly moving catalyst or trap. In the trap
case, i.e., $\gamma<0$, we look at the annealed time asymptotics in terms of
the first moment of $u$. Given a localized initial condition, we derive the
asymptotic rate of decay to zero in dimensions 1 and 2 up to equivalence and
characterize the limit in dimensions 3 and higher in terms of the Green's
function of a random walk. For a homogeneous initial condition we give a
characterisation of the limit in dimension 1 and show that the moments remain
constant for all time in dimensions 2 and higher. In the case of a moving
catalyst ($\gamma>0$), we consider the solution $u$ from the perspective of the
catalyst, i.e., the expression $u(t,Y_t+x)$. Focusing on the cases where
moments grow exponentially fast (that is, $\gamma$ sufficiently large), we
describe the moment asymptotics of the expression above up to equivalence.
Here, it is crucial to prove the existence of a principal eigenfunction of the
corresponding Hamilton operator. While this is well-established for the first
moment, we have found an extension to higher moments.
