 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 $u(x)={\mathbb{E}}_x[f(B_{{\tau }_D})]$.