## The Annals of Probability

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

Xia Chen

#### 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.

#### Article information

Source
Ann. Probab., Volume 40, Number 4 (2012), 1436-1482.

Dates
First available in Project Euclid: 4 July 2012

https://projecteuclid.org/euclid.aop/1341401141

Digital Object Identifier
doi:10.1214/11-AOP655

Mathematical Reviews number (MathSciNet)
MR2978130

Zentralblatt MATH identifier
1259.60094

#### Citation

Chen, Xia. Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models. Ann. Probab. 40 (2012), no. 4, 1436--1482. doi:10.1214/11-AOP655. https://projecteuclid.org/euclid.aop/1341401141

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