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\math bb{E}_0\exp\biggl{\pmθ\int_0^t\bar{V}(B_s) ds\biggr}\qquad (t\to\infty)] are investigated in connection with the renormalized Poisson potential of the form [\bar{V}(x)=\int_{\mathbb{R}^d}{\frac{1}{|y-x|^p}}[ω(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.

Published in at http://dx.doi.org/10.1214/11-AOP655 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials
• Quenched law of large numbers for branching Brownian motion in a random medium