Let $\sigma$ be the first time Brownian motion started outside the unit half-disc reaches its semicircular boundary. A path that reaches $A$ must first cross that semicircle. The at $\sigma$ gives
$$
\mathbb E_z[\operatorname{Im}B_\tau]
=\int_0^\pi p(z,e^{i\theta})
\mathbb E_{e^{i\theta}}[\operatorname{Im}B_\tau]\,d\theta.
$$
Take $z=iy$, multiply by $y$, and let $y\to\infty$. The <Brownian representation of half-plane capacity> identifies the left side with $\operatorname{hcap}(A)$, while part b gives
$$
yp(iy,e^{i\theta})\longrightarrow\frac2\pi\sin\theta.
$$
Since the exit height is between zero and one, the <dominated convergence theorem> applies and yields
$$
\operatorname{hcap}(A)
=\frac2\pi\int_0^\pi
\mathbb E_{e^{i\theta}}[\operatorname{Im}B_\tau]
\sin\theta\,d\theta.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-203/1/c/solution