Mapping-out function of a compact H-hull ($g_A$)

The mapping-out function is the unique conformal map $g_A:\mathbb{H}\setminus A\to \mathbb{H}$ with hydrodynamic normalization at infinity

A conformal map from an upper-half-plane domain has hydrodynamic normalization when $g(z)-z\to 0$ at infinity. For a compact H-hull this normalization makes its mapping-out function of a compact H-hull unique.

The half-plane capacity of a compact H-hull is the nonnegative coefficient $a$ in

For $\lambda >0$ and $x\in \mathbb{R}$,

If compact H-hulls satisfy $A\subseteq C$, thenThis follows from the Brownian representation of half-plane capacity or from composition of their mapping-out functions.

If $\tau$ is the first exit time of planar Brownian motion from $\mathbb{H}\setminus A$, thenIt follows by applying the optional sampling theorem for a supermartingale to the harmonic function $\mathrm{Im}(z-g_A(z))$.

There is a universal constant $C$ such that every compact H-hull satisfiesTranslate and scale so the hull lies in a unit half-disc, then use the Brownian representation of half-plane capacity and the $O(y^{-1})$ harmonic measure of that half-disc as viewed from $iy$.

For $R_r=[-r,r]\times (0,1]$ with $r\ge 1$,Indeed, the imaginary part at the Brownian exit point is at most one and the probability of reaching the radius-$O(r)$ neighbourhood containing the rectangle from $iy$ is $O(r/y)$. Consequently $r^{-1}{R}_r=[-1,1]\times (0,r^{-1}]$ has capacity $O(r^{-1})$ although its diameter tends to two.

A growing hull is parameterized by half-plane capacity when $\mathrm{hcap}(K_t)=2t$. The factor two makes its Chordal Loewner equation take the conventional form ${\partial }_t{g}_t=2/(g_t-U_t)$.