Quenched asymptotics for Brownian motion in generalized Gaussian potential

Abstract

In this paper, we study the long-term asymptotics for the quenched moment

\[\mathbb{E}_{x}\exp \biggl\{\int_{0}^{t}V(B_{s})\,ds\biggr\}\]

consisting of a $d$-dimensional Brownian motion $\{B_{s};s\ge0\}$ 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.