  In this paper, we study the long-term asymptotics for the quenched moment
\[\mathbb{E}_x\exp \biggl\{\int_0^tV(B_s)\,ds\biggr\}\] consisting of a
$d$-dimensional Brownian motion $\{B_s;s\ge 0\}$ and a generalized Gaussian
field $V$. The major progress made in this paper includes: Solution to an open
problem posted by Carmona and Molchanov [Probab. Theory Related Fields 102
(1995) 433-453], the quenched laws for Brownian motions in Newtonian-type
potentials and in the potentials driven by white noise or by fractional white
noise.
