High-level escape representation of a mapping-out height ($\mathrm{Im}{g}_K(z)={\lim }_{R\to \infty }R{\mathbb{P}}_z(\mathrm{Im}{B}_{{\tau }_R}=R)$)

Let $D=\mathbb{H}\setminus K$ and stop planar Brownian motion from $z\in D$ on reaching height $R$ or leaving $D$. Then $\mathrm{Im}{g}_K(z)={\lim }_{R\to \infty }R{\mathbb{P}}_z(\mathrm{Im}{B}_{{\tau }_R}=R)$. The stopped harmonic height is a bounded martingale. The bounded difference $g_K(z)-z$ makes its top-boundary value differ from $R$ by a uniform constant, while optional stopping of the imaginary coordinate bounds the top exit probability by $\mathrm{Im}z/R$. Killing on the hull is essential.