= Brownian hitting probability of a ball
{c}
{title2=$\mathbb P_x(T_\varepsilon<\infty)=(\varepsilon/|x|)^{d-2}$}
For $d\geq3$ and $|x|>\varepsilon$, stop the <harmonic function> $u(x)=|x|^{2-d}$ at the two boundaries of $\{\varepsilon<|x|<R\}$. Bounded <optional stopping> gives the probability of hitting the inner sphere first as $(u(x)-u(R))/(u(\varepsilon)-u(R))$. Letting $R\to\infty$ gives the displayed formula; a finite hitting path is contained in some finite-radius ball.
Back to article page
