Annealed asymptotics for Brownian motion of renormalized potential in mobile random medium

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 $t>0$.