Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 6 / a / Solution

For Brownian motion started at $x\in D$, the Itô formula and $\Delta u=0$ show thatis a local martingale. Since $D$ is bounded and $u$ is continuous on the compact set $\overline{D}$, the stopped process is bounded and hence a true martingale. Brownian motion exits every bounded domain almost surely, so $t\wedge \tau \to \tau$. The dominated convergence theorem, continuity at the boundary, and $u=f$ on $\partial D$ give the Brownian representation of the Dirichlet problem