= Brownian exit-time ball averaging identity
{c}
{title2=$g_D(x)=r^2/(n+2)+|B(x,r)|^{-1}\int_{B(x,r)}g_D(y)\,dy$}
Let $g_D(x)=\mathbb E_xT_D$, 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 $ns^{n-1}/r^n$ gives $r^2/(n+2)$. The identity gives local integrability whenever $g_D(x)$ is finite.
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