 Harmonic capacity from infinity in the upper half-plane

This capacity measures the asymptotic probability that planar Brownian motion hits a compact H-hull before the real axis. It scales linearly with length and is distinct from half-plane capacity. A vertical slit of height $h$ has capacity $2h$; a half-disc of radius $r$ has capacity $4r$. Monotonicity of Brownian hitting probabilities gives $\mathrm{cap}(K)\le 4\mathrm{rad}(K)$, where the radius is that of the least real-centred enclosing half-disc. For finite slit hulls the capacity is the length of the image of the two-sided absorbing boundary under the mapping-out function.