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Let $g_D(x)={\mathbb{E}}_x{T}_D$, possibly infinite, and suppose the closed ball $\overline{B(x,r)}$ lies in the open set $D$. A Strong Markov property decomposition at an independent random-radius exit time, using uniform ball sampling by Brownian stopping, gives the displayed identity in the nonnegative extended reals. Indeed the expected exit time from a radius-$s$ ball started at its centre is $s^2/n$, and averaging this against $n{s}^{n-1}/r^n$ gives $r^2/(n+2)$. The identity gives local integrability whenever $g_D(x)$ is finite.

Let $R=|U|$. Uniform volume measure on the unit ball gives $\mathbb{P}(R\le r)=r^n$, $0\le r\le 1$, hence radius density $n{r}^{n-1}$. Conditional on $R=r>0$, independence leaves $W$ a standard Brownian motion. Its exit time from the radius-$r$ ball is finite: applying optional stopping to $|W_t{|}^2-nt$ stopped at that exit and then letting $t\to \infty$ gives $\mathbb{E}{\tau }_r=r^2/n$.

The probability law of the exit point is invariant under every orthogonal transformation, because both Brownian motion and the centred sphere are invariant under those transformations. The unique invariant probability on that sphere is normalized surface measure; for $n=1$ it is the equally weighted pair $\{-r,r\}$. This is also the conditional angular law of $U$ given $|U|=r$. Mixing these identical conditional laws against $n{r}^{n-1}dr$ proves the uniform ball sampling by Brownian stopping identityThe event $R=0$ has probability zero and causes no difficulty. Independence of $R$ permits its value to be included in the initial filtration, making $T$ a stopping time.