For Brownian motion started at $x\in D$, the <Itô formula> and $\Delta u=0$ show that
$$
u(B_{t\wedge\tau})
$$
is 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>
$$
u(x)=\mathbb E_xu(B_{t\wedge\tau})
\longrightarrow\mathbb E_xu(B_\tau)
=\mathbb E_xf(B_\tau).
$$

Solution

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

{c}

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, so
$$
u(x)=\mathbb E_x[u(B_{\tau_D})].
$$
If $u=f$ on the boundary, this becomes $u(x)=\mathbb E_x[f(B_{\tau_D})]$.

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 Incoming links: Brownian representation of the Dirichlet problem