= Restriction probability of a Brownian half-plane excursion
{title2=$\mathbb P(\widehat B\cap A=\varnothing)=\psi_A'(0)$}
For a <compact H-hull> avoiding zero, let $\psi_A$ map its complement to the half-plane, fix zero, and have derivative one at infinity. The <Brownian excursion in the upper half-plane> avoids $A$ with probability $\psi_A'(0)$. From an interior starting point $z$, the corresponding transformed process avoids $A$ with probability $\operatorname{Im}g_A(z)/\operatorname{Im}z$; taking $z=i\varepsilon$ and $\varepsilon\downarrow0$ gives the boundary derivative.
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