= Bounded harmonic representation with almost sure Brownian exit
{title2=$u(z)=\mathbb E_z[u(B_{T_D})]$}
= Brownian representation with bounded harmonic values and finite exit time
{c}
{synonym}
If $u$ is a bounded <harmonic function> on a domain $D$, continuous on its closure, and the <Brownian exit time> $T_D$ is almost surely finite from $z$, then $u(z)=\mathbb E_z[u(B_{T_D})]$. The <Itô formula> and compact localization make $u(B_{t\wedge T_D})$ a bounded <martingale>. <bounded convergence theorem> at the exit time proves the representation even if the domain is unbounded. Removing boundedness of $u$ can invalidate this conclusion through loss of <uniform integrability>.
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