Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models

Abstract

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

\[\log\mathbb{E}_{0}\exp\left\{\pm\theta\int_{0}^{t}\overline{V}(B_{s})\,ds\right\}\qquad (t\to\infty)\]

are investigated in connection with the renormalized Poisson potential of the form

\[\overline{V}(x)=\int_{\mathbb{R}^{d}}{\frac{1}{\vert y-x\vert^{p}}}[\omega(dy)-dy],\qquad x\in\mathbb{R}^{d}.\]

The investigation is motivated by some practical problems arising from the models of Brownian motion in random media and from the parabolic Anderson models.