Fix a closed ball $\overline B(x,r)\subset D$ and let $\tau$ be its first exit time. By the , 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 $x$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 4 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-201/4/c/solution