= Brownian sphere-hitting probability in dimension three
{c}
For standard three-dimensional <Brownian motion> started at $x$ with $r<|x|<R$,
$$
\mathbb P_x(\tau_r<\tau_R)=\frac{|x|^{-1}-R^{-1}}{r^{-1}-R^{-1}}.
$$
The <harmonic function> $x\mapsto1/|x|$ gives a bounded <stopped martingale> in the annulus, so the <optional stopping theorem> gives the formula. Taking $R\to\infty$ yields $\mathbb P_x(\tau_r<\infty)=r/|x|$ for $|x|>r$. This limit uses path continuity on the finite interval before a hit, rather than assuming transience to prove transience.
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