= High-level escape representation of a mapping-out height
{title2=$\operatorname{Im}g_K(z)=\lim_{R\to\infty}R\mathbb P_z(\operatorname{Im}B_{\tau_R}=R)$}
Let $D=\mathbb H\setminus K$ and stop <planar Brownian motion> from $z\in D$ on reaching height $R$ or leaving $D$. Then $\operatorname{Im}g_K(z)=\lim_{R\to\infty}R\mathbb P_z(\operatorname{Im}B_{\tau_R}=R)$. 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 $\operatorname{Im}z/R$. Killing on the hull is essential.
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