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

TL;DR

This paper studies the long-term behavior of the logarithmic moment generating function for Brownian motion in a renormalized Poisson potential, relevant to models of random media and the parabolic Anderson model.

Contribution

It provides almost sure asymptotics for the logarithmic moment generating function in the context of renormalized Poisson potentials, extending understanding of Brownian motion in random media.

Findings

Derived asymptotic formulas for the logarithmic moment generating function.

Connected results to models of Brownian motion in random media.

Provided insights into the parabolic Anderson model behavior.

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{\pm\theta\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}}[\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.