 Boundary derivative is an excursion avoidance probability

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 $\mathrm{Im}{g}_A(z)/\mathrm{Im}z$. Letting $z=x+i\epsilon$ approach the boundary and using the Schwarz reflection principle gives the derivative. In particular $0<g_A^{\prime }(x)\le 1$ at every regular real boundary point away from the hull.