Small-ball Brownian hitting asymptotic (${\lim }_{\epsilon \downarrow 0}{\epsilon }^{2-d}{\mathbb{P}}_x(T_{\epsilon }\le t)=A_d^{-1}{\int }_0^t{p}_s(x)ds$)

For $d\ge 3$, fixed $x\ne 0$ and $t>0$, apply optional stopping to $u(B_{t\wedge T_{\epsilon }})$ with $u(z)=|z{|}^{2-d}$. The resulting identity is ${\epsilon }^{2-d}{\mathbb{P}}_x(T_{\epsilon }\le t)=u(x)-\mathbb{E}u(B_t)+\mathbb{E}[u(B_t);T_{\epsilon }\le t]$. The last term tends to zero by integrability of $u(B_t)$ and the vanishing total hitting probability. The Newtonian potential of the Brownian heat kernel identifies the remaining difference.