The problem is: consider A a solid ball centered at 0 and the exterior starting point $x\in A^{c}$, what is the behavior of $P\_{x}(T\_{B\_{r}(0)}>t)$ for $d\geq 3$ as $t\to \infty$,where $T\_{B\_{r}(0)}=inf\_{t>0}(B(t)\in B\_{r}(0))$?


In the literature, I find the joint distibution of $T\_{A}$ and $B(T\_{A})$ in terms of transforms that I just started learning about ("Hitting spheres from the exterior" by Betz and Gzyl). So I hope someone can help me translate it to what it means for $P\_{x}(T\_{B\_{r}(0)}>t)$.


Also,if you know the answer, please tell me which methods you used eg. Wiener sausage, heat equation, Feynman-Kac, Borel-Cantelli etc.


Just to be clear I was not looking for the above classic result, but instead a decaying estimate for $P\_{x}(T\_{B\_{r}(0)}>t)$ eg. $P\_{x}(T\_{B\_{r}(0)}>t)\leq e^{-rt}+(\frac{r}{x})^{d/2}$.


Thank you



Asymptotics for Hitting the sphere from the Outside
In "Heat flow, Brownian motion and Newtonian capacity" by Van den Berg


We have $P\_{x}[t<T\_{B\_{0,r}}<\infty]=\int\_{t}^{\infty}(\frac{1}{4\pi s^{3}})^{1/2}\frac{r(|x|-r)}{|x|}e^{-\frac{(|x|-r)^{2}}{4s}}ds$


