 Solution

Write $u(z)=|z{|}^{2-d}$. For a radial function $r^{2-d}$, the Laplacian isThus $u$ is a harmonic function off zero. Fix $R>|x|$ and stop Brownian motion when it reaches either sphere of radius $\epsilon$ or $R$, at time $S_R$. This exit time is finite: stopping the martingale $|B_t{|}^2-dt$ at $S_R\wedge t$ yields $d\mathbb{E}(S_R\wedge t)\le R^2-|x{|}^2$, and monotone convergence applies. By Itô's formula, $u(B_{t\wedge S_R})$ is a bounded martingale. Optional stopping and dominated convergence therefore giveThe events $\{T_{\epsilon }<T_R\}$ increase to $\{T_{\epsilon }<\infty \}$ as $R\to \infty$: a continuous path up to any finite hitting time is bounded. Since $d\ge 3$, $R^{2-d}\to 0$, proving the Brownian hitting probability of a ball