= Small-ball Brownian hitting asymptotic
{title2=$\lim_{\varepsilon\downarrow0}\varepsilon^{2-d}\mathbb P_x(T_\varepsilon\leq t)=A_d^{-1}\int_0^tp_s(x)\,ds$}
For $d\geq3$, fixed $x\ne0$ and $t>0$, apply <optional stopping> to $u(B_{t\wedge T_\varepsilon})$ with $u(z)=|z|^{2-d}$. The resulting identity is $\varepsilon^{2-d}\mathbb P_x(T_\varepsilon\leq t)=u(x)-\mathbb Eu(B_t)+\mathbb E[u(B_t);T_\varepsilon\leq t]$. 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.
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