  Let $B_s$ be a $d$-dimensional Brownian motion and $\omega(dx)$ be an
independent Poisson field on $\mathbb{R}^d$. The almost sure asymptotics for
the logarithmic moment generating function [\log\math
bb{E}_0\exp\biggl{\pm\theta\int_0^t\bar{V}(B_s) ds\biggr}\qquad (t\to\infty)]
are investigated in connection with the renormalized Poisson potential of the
form [\bar{V}(x)=\int_{\mathbb{R}^d}{\frac{1}{|y-x|^p}}[\omega(dy)-dy],\qquad
x\in\mathbb{R}^d.] The investigation is motivated by some practical problems
arising from the models of Brownian motion in random media and from the
parabolic Anderson models.
