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For the Locality property of SLE, take a compact H-hull $A$ away from $0$, and map $\mathbb{H}\setminus A$ conformally to $\mathbb{H}$ by ${\psi }_A$, fixing $0$ and infinity and with derivative one at infinity. Up to the first contact with $A$, the image under ${\psi }_A$ of an ${\mathrm{SLE}}_6$ grown in $\mathbb{H}$ has the same law as ${\mathrm{SLE}}_6$ grown in $\mathbb{H}$, after the change to its own half-plane-capacity parameterization. Equivalently, ${\mathrm{SLE}}_6$ in the smaller domain has the same initial law as the curve in the larger domain until the latter encounters the removed hull. There is no conditioning on eventual avoidance.

The chordal restriction property for ${\mathrm{SLE}}_{8/3}$ concerns the whole curve. Conditional on avoiding $A$, applying ${\psi }_A$ gives the original ${\mathrm{SLE}}_{8/3}$ law, modulo increasing reparameterization:Its associated avoidance probability, from the SLE eight-thirds restriction martingale, isThus locality identifies a stopped, unconditioned law, whereas restriction identifies a conditioned law of the entire curve.

A compact H-hull is a bounded, relatively closed set $A\subset \mathbb{H}$ for which $\mathbb{H}\setminus A$ is simply connected domain. Its mapping-out function of a compact H-hull is the unique conformal map $g_A:\mathbb{H}\setminus A\to \mathbb{H}$ with hydrodynamic normalization at infinityThe half-plane capacity is

The statement is true. In the notation of part (ii), equality of the capacities forces $\mathrm{hcap}(D)=0$. Every nonempty compact H-hull has strictly positive half-plane capacity: by the Brownian representation of half-plane capacity, Brownian motion started sufficiently high has positive harmonic measure of a boundary portion of positive height. Hence $D=\varnothing$, so $C\setminus A=\varnothing$ and

Map out $A$ first. The imagewith its bounded filling is a compact H-hull, and uniqueness of hydrodynamic normalization givesComparing the coefficients of $1/z$ at infinity yields the half-plane-capacity composition ruleThus half-plane capacity is monotone under inclusion.

A compact H-hull is a bounded relatively closed set $A\subset \mathbb{H}$ for which $\mathbb{H}\setminus A$ is a simply connected domain. Its mapping-out function of a compact H-hull is the unique conformal map $g_A:\mathbb{H}\setminus A\to \mathbb{H}$ with hydrodynamic normalization at infinityIts half-plane capacity is $\mathrm{hcap}(A)=a$.

False. LetAfter filling bounded complementary components if necessary, this is a compact H-hull with diameter asymptotic to $2n$. The half-plane capacity of a low rectangle estimate, together with scaling and translation of half-plane capacity, givesThus unbounded diameter need not force unbounded capacity.

A compact H-hull is a bounded, relatively closed set $A\subset \mathbb{H}$ such that $\mathbb{H}\setminus A$ is simply connected. Its mapping-out function of a compact H-hull is the unique conformal bijectionwith hydrodynamic normalization

A compact H-hull is a bounded relatively closed set $A\subset \mathbb{H}$ such that $\mathbb{H}\setminus A$ is a simply connected domain. Its mapping-out function of a compact H-hull is the unique conformal map $g_A:\mathbb{H}\setminus A\to \mathbb{H}$ with hydrodynamic normalization at infinityThe nonnegative coefficient $a$ is the half-plane capacity $\mathrm{hcap}(A)$.

A compact H-hull is a bounded relatively closed set $A\subset \mathbb{H}$ for which $\mathbb{H}\setminus A$ is a simply connected domain. The Riemann mapping theorem and hydrodynamic normalization at infinity give a unique mapping-out function of a compact H-hull $g_A:\mathbb{H}\setminus A\to \mathbb{H}$ withThe coefficient $a\ge 0$ is the half-plane capacity $\mathrm{hcap}(A)$.

For $\lambda >0$ and $x\in \mathbb{R}$, uniqueness of the normalized map givesSubstituting the expansion of $g_A$ yieldsTherefore the scaling and translation of half-plane capacity is

For $c=\sqrt{1-{\epsilon }^2}$, three vertical slits of height $\epsilon$ rooted at $-c,0,c$ form a compact H-hull of enclosing radius one. Subadditivity of harmonic hull capacity gives capacity at most $6\epsilon$, which can tend to zero. Thus a uniform lower bound requires additional geometric hypotheses.

If a compact H-hull lies in the unit disc, compare its tail harmonic measure with that of the empty hull and filled half-disc. The Poisson kernel for the upper half-plane turns the inclusion of Brownian exit events into $x\le g_K(x)\le x+1/x$ for $x>1$. Reflection gives $x+1/x\le g_K(x)\le x$ for $x<-1$.

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 ${\psi }_A^{\prime }(0)$. From an interior starting point $z$, the corresponding transformed process avoids $A$ with probability $\mathrm{Im}{g}_A(z)/\mathrm{Im}z$; taking $z=i\epsilon$ and $\epsilon \downarrow 0$ gives the boundary derivative.

A holomorphic self-map of the unit disc contracts its pseudohyperbolic distance. Conjugating by Möbius transformations that send one chosen source point and its image to zero reduces the statement to the Schwarz lemma. In the complex upper half-plane this givesIf $f(iY)=iY+o(Y)$ at infinity, multiply the equivalent inequality between one minus the squared distances by $Y$ and let $Y\to \infty$. It yields $\mathrm{Im}f(z)\ge \mathrm{Im}z$. Applied to the inverse mapping-out function of a compact H-hull, this proves the nonnegativity of its half-plane capacity.

For a compact H-hull inside the disc of radius $r$ centred at a real point, its mapping-out function of a compact H-hull displaces every point of its domain by at most $3r$. After scaling and translation, the real boundary bounds for a unit-disc H-hull put the corresponding inverse boundary interval inside $[-2,2]$. Boundary cluster points of the inverse over this interval lie in the unit disc. The maximum modulus principle applied to $w-g_K^{-1}(w)$ gives the bound without assuming local connectedness of the hull. The nearly closed semicircular slit makes the constant three sharp.

When a finite union is a compact H-hull, hitting that union before the real axis implies hitting at least one member before the real axis. The union bound followed by the defining limit proves subadditivity of harmonic capacity from infinity in the upper half-plane.

The chordal SLE Loewner trace eventually leaves every bounded subset of the complex upper half-plane. For $0<\kappa <4$, its time-one compact H-hull is a simple slit and does not contain a half-disc. Instead, the boundary approach forces a small SLE Bessel gap estimate shows that every fixed nonzero real point is outside the closure of the full Loewner trace. After mapping out the initial simple slit, the starting point has two distinct real prime end images. A fresh independent SLE avoids approaching either image, so the original future stays a positive distance from its starting point. Scaling invariance of SLE gives ${\inf }_{t\ge T}|\gamma (t)|\stackrel{d}{=}\sqrt{T}{\inf }_{t\ge 1}|\gamma (t)|$, and decreasing-event continuity proves escape from every fixed disc. The separate argument based on a swallowed half-disc at positive time applies in the nonsimple regime $\kappa >4$; it must not be applied to a simple slit. The critical parameters require their own limiting or trace-existence arguments.