= Boundary derivative is an excursion avoidance probability
{title2=$g_A'(x)=\mathbb P_x^{\rm exc}(\widehat B\cap A=\varnothing)$}
For a <compact H-hull> $A$ whose closure avoids a real boundary point $x$, the <Brownian excursion in the upper half-plane> from $x$ to infinity avoids $A$ with the displayed probability. From an interior point $z$, the corresponding <Doob h-transform> avoidance probability is $\operatorname{Im}g_A(z)/\operatorname{Im}z$. Letting $z=x+i\varepsilon$ approach the boundary and using the <Schwarz reflection principle> gives the derivative. In particular $0<g_A'(x)\leq1$ at every regular real boundary point away from the hull.
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