Quenched asymptotics for Brownian motion in generalized Gaussian potential

Abstract

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.