Solution
= Solution
The function $u(x)=\epsilon/|x|$ is <harmonic function>[harmonic] on $\{|x|>\epsilon\}$, equals one on the target sphere, and tends to zero at infinity. Optional stopping at the first hit of radius $\epsilon$ and the first exit from a ball of radius $R$ gives
$$
\mathbb P_x(\tau_\epsilon<\tau_R)
=\frac{|x|^{-1}-R^{-1}}{\epsilon^{-1}-R^{-1}}.
$$
Letting $R\to\infty$ and taking $|x|=1$ yields $\mathbb P_x(\tau_\epsilon<\infty)=\epsilon$.