Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 34 / 4

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

To keep the normalization visible, denote the integral defining the printed constant by $A_d$. Substituting $r=|y{|}^2/(2s)$ gives, for $y\ne 0$,This is the Newtonian potential of the Brownian heat kernel. The singularity at zero is locally integrable in dimension $d$, since its radial integral behaves like ${\int }_0^1{r}^{2-d}{r}^{d-1}dr=1/2$. All integrands are nonnegative, so Tonelli's theorem permits both orders of integration.

Integrating in $s$ first givesIntegrating in $y$ first uses the Gaussian heat kernel convolution. Completing the square in $|y-x{|}^2/t+|y{|}^2/s$ givesConsequently $I={\int }_t^{\infty }p(r,0,x)dr$, and the actual identity isThe original PDF uses $A_d$ rather than $A_d^{-1}$ as the multiplier while defining its constant to equal $A_d$. That formula is false. For example $A_3=1/(2\pi )$, so the required multiplier in dimension three is $2\pi$. Equivalently, as $t\downarrow 0$ at $x\ne 0$, the left side tends to $|x{|}^{2-d}$, whereas the printed right side tends to $A_d^2|x{|}^{2-d}$. One can retain the printed form only by defining its multiplier as $c_d=A_d^{-1}$.

Continue with $A_d$ equal to the integral defined in the PDF and $u(z)=|z{|}^{2-d}$. Until the Brownian motion hits the radius-$\epsilon$ ball, $u(B_s)$ is a bounded local martingale, hence a true martingale. At the finite stopping time $t\wedge T_{\epsilon }$, optional stopping givesRearranging,The fixed random variable $u(B_t)$ is integrable by the Gaussian heat kernel and the local-integrability calculation above. Also ${\mathbb{P}}_x(T_{\epsilon }\le t)\le (\epsilon /|x|{)}^{d-2}\to 0$. For any $R>0$, the final expectation is at mostFirst send $\epsilon \downarrow 0$, then $R\to \infty$, proving that it vanishes. The identities from part (b) giveSubtracting yields the small-ball Brownian hitting asymptoticThus the constant in this printed limit requires exactly the same reciprocal correction as in part (b). For $d=3$ the multiplier is $2\pi$, not $1/(2\pi )$. The argument proves the corrected finite-time limit directly, rather than inferring it from the eventual hitting probability alone.