The Annals of Probability

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

Xia Chen

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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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1341401141

Digital Object Identifier
doi:10.1214/11-AOP655

Mathematical Reviews number (MathSciNet)
MR2978130

Zentralblatt MATH identifier
1259.60094

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60K37: Processes in random environments 60K40: Other physical applications of random processes 60G55: Point processes 60F10: Large deviations

Keywords
Renormalization Poisson field Brownian motion in Poisson potential parabolic Anderson model Feynman–Kac representation large deviations

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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