Asymptotics for Hitting the sphere from the Outside

Question

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

Answer 1

The quantity you asked about converges to $1-u(x):=P^x(T_{B_r(0)}=\infty)$; $u(x)$ is a harmonic function which equals $1$ on the boundary of the ball and equals $0$ at infinity; Thus, $u(x)=(r/|x|)^{d-2}$.

Answer 2

In "Heat flow, Brownian motion and Newtonian capacity" by Van den Berg

We have $P_{x}[t<T_{B_{0,r}}<\infty]=\int_{t}^{\infty}(\frac{1}{4\pi s^{3}})^{1/2}\frac{r(|x|-r)}{|x|}e^{-\frac{(|x|-r)^{2}}{4s}}ds$