Spatial Brownian motion in renormalized Poisson potential: A critical case

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{\theta \int_0^t \bar{V}(B_s) ds} \ < \infty
a.s. & $\text{if}$ \theta< 1/16, \medskip = \infty \ a.s. & $\text{if}$ \theta> 1/16, where $\bar{V}(x)$ is the renormalized Poisson potential $$\bar{V}(x)={\int }_{{\mathbb{R}}^3}\frac{1}{\mathrm{\mid }x-y{\mathrm{\mid }}^2}[\omega (dy)-dy]\mathrm{.}$$ Then the long term behavior of the quenched exponential moment \eqref{} is determined for $\theta \in (0,1\mathrm{/}16)$ 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 $${\int }_{{\mathbb{R}}^3}\frac{f^2(x)}{\mathrm{\mid }x{\mathrm{\mid }}^2}dx\le 4\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_2^2,2inf\in W^{1,2}({\mathbb{R}}^3)\mathrm{.}$$