Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 220 / 3 / b / Solution

Set $h_A(z)=\mathrm{Im}(z-g_A(z))$. This is a nonnegative harmonic function on $\mathbb{H}\setminus A$, has boundary value $\mathrm{Im}z$ on the hull boundary, and tends to zero on the real boundary. If $\tau$ is the first exit time of planar Brownian motion from $\mathbb{H}\setminus A$, the optional stopping theorem givesThe hydrodynamic expansion yields $h_A(iy)=\mathrm{hcap}(A)/y+O(y^{-2})$, and therefore the Brownian representation of half-plane capacity