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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 Gaussianfield $V$. The major progress made in this paper includes: Solution to an openproblem posted by Carmona and Molchanov [Probab. Theory Related Fields 102(1995) 433-453], the quenched laws for Brownian motions in Newtonian-typepotentials and in the potentials driven by white noise or by fractional whitenoise.

Quenched asymptotics for Brownian motion in generalized Gaussian potential