Brownian representation of the Dirichlet problem

Let $D$ be bounded, let $u$ be harmonic on $D$ and continuous on $\overline{D}$, and let ${\tau }_D$ be the first exit time of Brownian motion. The Itô formula makes $u(B_{t\wedge {\tau }_D})$ a bounded martingale, soIf $u=f$ on the boundary, this becomes $u(x)={\mathbb{E}}_x[f(B_{{\tau }_D})]$.

Almost-sure finiteness of the exit time does not replace boundedness of the domain. In the upper half-plane, $u(x,y)=y$ is harmonic with zero boundary values and Brownian motion reaches the boundary almost surely, but $u$ is positive inside. The stopped martingale fails to be uniformly integrable.