Spatial Brownian motion in renormalized Poisson potential: A critical case

Abstract

Let $B_s$ be a three dimensional Brownian motion and $\omega (dx)$ be an independent Poisson field on ${\mathbb{R}}^3$. It is proved that for any $t>0$, conditionally on $\omega (\cdot )$, \label{*} \mathbb{E}_0 \exp\{θ\int_0^t \bar{V}(B_s) ds\} \ < \infty \ a.s. & \text{if} θ< 1/16, \medskip = \infty \ a.s. & \text{if} θ> 1/16, where $\bar{V}(x)$ is the renormalized Poisson potential $$\bar{V}(x)={\int }_{{\mathbb{R}}^3}\frac{1}{|x-y{|}^2}[\omega (dy)-dy].$$ Then the long term behavior of the quenched exponential moment \eqref{*} is determined for $\theta \in (0,1/16)$ in the form of integral tests. This paper exhibits and builds upon the interrelation between the exponential moment \eqref{*} and the celebrated Hardy's inequality $${\int }_{{\mathbb{R}}^3}\frac{f^2(x)}{|x{|}^2}dx\le 4\Vert \nabla f{\Vert }_2^2,2inf\in W^{1,2}({\mathbb{R}}^3).$$