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July 2012 Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models
Xia Chen
Ann. Probab. 40(4): 1436-1482 (July 2012). DOI: 10.1214/11-AOP655

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.

Citation

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Xia Chen. "Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models." Ann. Probab. 40 (4) 1436 - 1482, July 2012. https://doi.org/10.1214/11-AOP655

Information

Published: July 2012
First available in Project Euclid: 4 July 2012

zbMATH: 1259.60094
MathSciNet: MR2978130
Digital Object Identifier: 10.1214/11-AOP655

Subjects:
Primary: 60F10, 60G55, 60J65, 60K37, 60K40

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 4 • July 2012
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