The problem is: Given compact set A with positive finite volume (eg. ball,cube), what happens to $P_{x}(T_{A}>t)$ as $t\to \infty$, where $T_{A}=inf_{t>0}(B_{t}\in A)$ and x is in the "exterior" of A (consider A to be a solid object, then x is in the complement).

More specifically, do we have for all finite t: $P_{x}(\infty>T_{A}>t)>0$ for $d\geq 3$?

Also, do we have decay estimates of $P_{x}(\infty>T_{A}>t)$?

In d=1,2, Brownian motion is recurrent and so $P_{x}(T_{A}>t)=0$ as $t\to \infty$. I want to know the behavior of $P_{x}(T_{A}>t)$ for $d\geq 3$ as $t\to \infty$. For example does it decay exponentially? Is it always positive?

What are the methods I can use to study this problem (eg. Wiener Sausage asymptotics, Feyman-Kac etc)?

I am also curious, how far we've gone in finding densities of stopping times for various shapes. For $T_{a}=inf_{t>0}(B_{t}=a)$, we have $P(T_{a}>t)=\int_{t}^{\infty}\frac{a}{\sqrt{2\pi(s-t)^{3}}}e^{\frac{-a^{2}}{2(s-t)}}ds$ and for the sphere $P_{x}[t<T_{B_{0,r}}<\infty]=\int_{t}^{\infty}(\frac{1}{4\pi s^{3}})^{1/2}\frac{r(|x|-r)}{|x|}e^{-\frac{(|x|-r)^{2}}{4s}}ds$

Any help will be appreciated.

Thanks