  Motivated by the study of the directed polymer model with mobile Poissonian
traps or catalysts and the stochastic parabolic Anderson model with time
dependent potential, we investigate the asymptotic behavior of
\[\mathbb{E}\otimes\mathbb{E}_0\exp\left\{\pm\theta\int^t_0\bar{V}(s,B_s)ds\right\}\qquad
(t\to\infty)\] where $\th>0$ is a constant, $\overline{V}$ is the renormalized
Poisson potential of the form
\[\overline{V}(s,x)=\int_{\mathbb{R}^d}\frac{1}{|y-x|^p}\left(\omega_s(dy)-dy\right),\]
and $\omega_s$ is the measure-valued process consisting of independent Brownian
particles whose initial positions form a Poisson random measure on
$\mathbb{R}^d$ with Lebesgue measure as its intensity. Different scaling limits
are obtained according to the parameter $p$ and dimension $d$. For the
logarithm of the negative exponential moment, the range of $\frac{d}{2}<p<d$ is
divided into 5 regions with various scaling rates of the orders $t^{d/p}$,
$t^{3/2}$, $t^{(4-d-2p)/2}$, $t\log t$ and $t$, respectively. For the positive
exponential moment, the limiting behavior is studied according to the
parameters $p$ and $d$ in three regions. In the sub-critical region ($p<2$),
the double logarithm of the exponential moment has a rate of $t$. In the
critical region ($p=2$), it has different behavior over two parts decided
according to the comparison of $\theta$ with the best constant in the Hardy
inequality. In the super-critical region $(p>2)$,
  the exponential moments become infinite for all $t>0$.
