= Solution
Fix a closed ball $\overline B(x,r)\subset D$ and let $\tau$ be its first exit time. By the <Strong Markov property>, conditioning at $\tau$ gives
$$
\phi(x)=\mathbb E_x[\phi(X_\tau)].
$$
The <orthogonal invariance of Brownian motion> implies that $X_\tau$ is uniformly distributed on the sphere $\partial B(x,r)$. Hence
$$
\phi(x)=\frac1{|\partial B(x,r)|}
\int_{\partial B(x,r)}\phi(y)\,dS(y).
$$
Thus $\phi$ has the <mean value property> on every ball compactly contained in $D$. Since $0\leq\phi\leq1$, the mean-value characterization of <harmonic functions> yields
$$
\boxed{\Delta\phi=0\quad\text{in }D.}
$$
This function is the <harmonic measure> of $A$ viewed from $x$.
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