Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 4 / c / 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 $\tau$ givesThe orthogonal invariance of Brownian motion implies that $X_{\tau }$ is uniformly distributed on the sphere $\partial B(x,r)$. HenceThus $\varphi$ has the mean value property on every ball compactly contained in $D$. Since $0\le \varphi \le 1$, the mean-value characterization of harmonic functions yieldsThis function is the harmonic measure of $A$ viewed from $x$.