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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.

Suppose a capacity-parameterized Loewner chain lies in a domain on which $\psi$ is a conformal map. For its image chain set ${\psi }_t={\tilde{g}}_t\circ \psi \circ g_t^{-1}$. Then the image Loewner driving function is ${\tilde{U}}_t={\psi }_t(U_t)$ andThe derivative is real by the Schwarz reflection principle. Cancellation of the pole in the differentiated conjugacy identity proves the squared-derivative rule.

Work in the prime end closure of the marked domain, so that its two conformal boundary points are distinct even if their physical boundary representatives coincide. A chordal filling is a closed connected set joining those points, meeting the conformal boundary nowhere else, and full relative to the two boundary arcs between them: complementary pockets that do not reach either boundary arc have been added. In the normalized complex upper half-plane, its two complementary sides are simply connected and abut the negative and positive real axes. For a proper boundary-to-boundary curve, chordal filling means adjoining all such trapped pockets to its range. Independent curves are combined by taking their union and then applying this same chordal filling operation.

An admissible marked subdomain $V\subset U$ contains neighborhoods of the two marked boundary points and has a simply connected component joining them. The restriction property says that, conditional on the random chordal filling $K$ being contained in $V$, its law is the chordal filling law for $(V,z_0,z_1)$. Equivalently, mapping the conditional chordal filling conformally back to $(U,z_0,z_1)$ recovers the original law. The contained-set events are understood away from the shared marked endpoints.

By transporting the family under conformal maps, it suffices to work in $(\mathbb{H},0,\infty )$. Let $A$ be an admissible compact H-hull whose closure avoids zero, and normalizeIn terms of the single normalized measure $\mu$, restriction is preciselyThe Schwarz reflection principle extends ${\Phi }_A$ analytically near zero, with ${\Phi }_A^{\prime }(0)>0$, so the stated conditioning events have positive probability for the two laws considered below. This is the chordal restriction property for filled sets. An unbounded marked subdomain can be approximated by such compact-hull removals; conformal maps transport the formulation to general domains.

We need a determination principle for filled sets, since a probability of one avoidance event alone would not establish equality of laws. Choose countably many thin polygonal compact H-hulls with rational data, attached to real boundary intervals away from zero. For each chordal filling, every point of its complement has a path within one of the two complementary sides to the appropriate real boundary interval. A sufficiently thin tube along such a path avoids the chordal filling and contains a neighborhood of that point. Hence the union of interiors of all avoided test tubes is exactly the complement.

Their joint avoidance probabilities are also determined by single-hull probabilities. For a finite union of test compact H-hulls, either the union separates zero from infinity, in which case a connected chordal filling cannot avoid it, or its unbounded complementary component contains both marked endpoints. In the latter case fill the union's bounded pockets to get an admissible compact H-hull $C$. A connected set joining zero to infinity and avoiding the union cannot enter one of these separated pockets, so it avoids $C$ as well. Thus joint avoidance is either impossible or is the avoidance event for $C$. The inclusion-exclusion principle gives every finite joint probability distribution of the tube-avoidance indicators, and these indicators determine the random complement and chordal filling. This proves avoidance probabilities determine a chordal filling law.

In the given avoidance formulas, the argument of the conformal map must be the retained open domain, rather than the random closed filling. We express that domain as $\mathbb{H}\setminus A$ to keep the two objects distinct. The formulas then imply for the two filled processesThe chordal filling operation does not change avoidance of an admissible compact H-hull: every piece of that compact H-hull is attached to the real boundary away from the endpoints, and so cannot be inside a trapped pocket while its generating curve avoids the compact H-hull. These laws are conformally invariant as unparameterized chordal filling laws. For SLE this follows from Question 1; for the Brownian half-plane excursion, Brownian scaling of its independent one- and three-dimensional coordinates shows that its unparameterized law is unchanged by positive dilations, and mapping the Brownian half-plane excursion to another marked domain gives the conformally natural law. Conformal invariance of planar Brownian motion describes the corresponding change of time, which does not alter the chordal filling.

Here is the conditional-law calculation for either exponent $\alpha \in \{5/8,1\}$. For another admissible compact H-hull $B$ in the image complex upper half-plane, setThe complement of $C$ is mapped to $\mathbb{H}$ by this composition, so $C$ is again admissible; both maps fix zero and have derivative one at infinity. The chain rule gives ${\Phi }_C^{\prime }(0)={\Phi }_B^{\prime }(0){\Phi }_A^{\prime }(0)$. ThereforeThese are exactly the original chordal filling's avoidance probabilities for every $B$. The determination principle proves equality of the conditional image law with the original law. Hence both generated chordal fillings have restriction, with exponents $5/8$ and $1$ respectively. This is a complete implication from the two allowed avoidance formulas; the restriction conclusion was not assumed.

For independent chordal fillings, avoidance of $A$ by their filled union is equivalent to avoidance by every component. Independence multiplies the probabilities, and thus adds their restriction exponents of a chordal filling. ConsequentlyThe determination principle applies once more, givingThe equality concerns the filled, unparameterized closed sets. The numerical reason is $8\cdot (5/8)=5\cdot 1=5$.

Write $D=\mathbb{H}\setminus K$. By definition of a compact H-hull, $D$ is a proper simply connected domain, and it agrees with the complex upper half-plane outside a sufficiently large disc. The Riemann mapping theorem supplies a conformal isomorphism $f:D\to \mathbb{H}$. Choose its boundary normalization so that the prime end at infinity maps to infinity. This prime end is unambiguous because the domain has an ordinary straight real boundary near infinity.

The Schwarz reflection principle, applied there in the coordinate $1/z$, extends $f$ across the real boundary near infinity. It has a simple pole there, with Laurent seriesThe pole is simple because the reflected map is locally conformal at infinity; its leading coefficient is positive because it maps the upper side to the upper side. Postcomposing with the half-plane automorphism $w\mapsto (w-b)/a$ givesThis is hydrodynamic normalization at infinity.

If $g_1,g_2$ both satisfy it, $g_2\circ g_1^{-1}$ is a conformal automorphism of the upper half-plane. The asymptotics force it to fix infinity, so it has the form $w\mapsto \alpha w+\beta$ with $\alpha >0$ and $\beta \in \mathbb{R}$. The same asymptotics force $\alpha =1$, $\beta =0$. Hence the normalized mapping-out function exists and is unique. The argument requires reflection only outside a large disc, not regularity of the hull's entire boundary.

At a real point $x$ outside the hull's closure, the Schwarz reflection principle makes the mapping-out function analytic across the boundary and $g_K^{\prime }(x)>0$. Use the course's Brownian excursion restriction identity: for an excursion from $x$ to infinity in the complex upper half-plane,One way to obtain this identity is to start its Doob h-transform at $x+i\epsilon$: its avoidance probability is $\mathrm{Im}{g}_K(x+i\epsilon )/\epsilon$, which tends to $g_K^{\prime }(x)$ by reflection. Thus the boundary derivative is an excursion avoidance probability, and in particular it is at most one.

Let $D$ be the filled unit half-disc. Since $K\subseteq D$, avoiding $D$ implies avoiding $K$. The monotonicity of boundary derivatives of mapping-out functions and part (b) give, for $|x|>1$,The upper bound is attained by the empty hull and the lower by the filled half-disc, so the comparison is sharp on either real tail.

Let $g_t$ map out the original filled hull and let $g_t^{\ast }$ map out the filled hull generated by $\Phi (\gamma [0,t])$. Both have hydrodynamic normalization at infinity. WriteAlthough the original $\Phi$ is specified only on $N$, this conjugacy is defined in the surviving neighborhood of the tip. Boundary preservation gives a real analytic extension there by the Schwarz reflection principle, with $a_t>0$. The identities below are first applied on compactly localized intervals before exit and then continued up to $T$.

Set $u(t)=\mathrm{hcap}(K_t^{\ast })/2$, the capacity clock of the image hull. The Chordal Loewner equation for each flow and the chain rule giveAnalyticity at the driving point cancels the coefficient of $(z-{\xi }_t{)}^{-1}$, forcing $\dot{u}(t)=a_t^2$. The constant term, computed as in Question 3, is then ${\partial }_t{h}_t({\xi }_t)=-3b_t$. For a general SLE driver $d{\xi }_t=\sqrt{\kappa }d{B}_t$, the Itô formula consequently gives the transformed SLE driving functionAt $\kappa =6$, this becomesThus $W$ is a continuous local martingale up to the exit time. If the source notation ${\xi }_t^{\ast }$ indexes the image driver by original time, it denotes $W_t$. If it uses the image's own half-plane-capacity time, it denotes $W_{t(u)}$, with $t(u)$ the inverse clock; that time-changed process is also a local martingale. This distinguishes the two clocks without changing the requested conclusion.

A compact H-hull is a bounded relatively closed set $K\subset \mathbb{H}$ whose complement $D=\mathbb{H}\setminus K$ is a simply connected domain. Equivalently one can retain its compact closure in $\overline{\mathbb{H}}$; real boundary points do not affect the mapping-out function. The Riemann mapping theorem and Schwarz reflection principle near infinity give a unique conformal map $g_K:D\to \mathbb{H}$ with hydrodynamic normalization at infinity. Its Laurent series isThe half-plane capacity is $\mathrm{hcap}(K)=a_K$. The missing constant term is part of the normalization, not a choice to be made again when comparing different hulls.

Here are the conformal map facts we use to justify positivity and comparison. The inverse $f_K:\mathbb{H}\to D\subset \mathbb{H}$ has expansion $f_K(w)=w-a_K/w+O(w^{-2})$. The upper-half-plane Schwarz-Pick theorem, applied to $w$ and $iY$ and then letting $Y\to \infty$, implies $\mathrm{Im}{f}_K(w)\ge \mathrm{Im}w$. Indeed the inequality between pseudohyperbolic distances givesand multiplying by $Y$ gives the height comparison in the limit. At $w=iY$, the inverse expansion reads $\mathrm{Im}{f}_K(iY)=Y+a_K/Y+O(Y^{-2})$, so $a_K\ge 0$.

For $r>0$, uniqueness of the mapping-out function givesConsequentlyTranslation by a real number $b$ similarly gives $g_{K+b}(z)=b+g_K(z-b)$ and leaves half-plane capacity unchanged.

If $K\subset A$ are compact H-hulls, their mapping-out functions compose as $g_A=g_L\circ g_K$, where $L$ is the filled image of $A\setminus K$. Comparing Laurent series yieldsThis proves monotonicity of half-plane capacity, rather than assuming it from inclusion alone. The filled half-disc $A=\{z\in \mathbb{H}:|z-b|\le R\}$ has mapping-out functionwhich takes its semicircular boundary to the real line and is a conformal bijection onto $\mathbb{H}$. Thus its half-plane capacity is $R^2$. Enclose $K$ in such a half-disc, compare capacities, and minimize over real centres and radii. This proves the sharp boundIf one defines the radius by an infimum, take $R$ strictly larger than that infimum and then let $R$ decrease; no attainment assumption is needed.

Write $D=\mathbb{H}\setminus K$. The mapping-out function is a conformal map $g_K:D\to \mathbb{H}$ with hydrodynamic normalization at infinityEquivalently, $g_K(z)-z\to 0$. The Riemann mapping theorem gives a map onto the half-plane. Since $K$ is bounded, infinity has an analytic real-boundary neighborhood. After sending its image to infinity, the Schwarz reflection principle gives a Laurent series $az+b+O(z^{-1})$ there, with $a>0$ and $b$ real. A real affine automorphism removes $a$ and $b$, giving the stated normalization. Reflection also makes $a_K$ real.

If $g_1,g_2$ both have this normalization, $g_2{g}_1^{-1}$ is a conformal automorphism of the upper half-plane. Such maps are real Möbius transformations. The expansion forces it to fix infinity, have leading coefficient $1$, and have zero translation. It is therefore the identity. Thus the normalized mapping-out function exists and is unique.

The mapping-out function of a compact H-hull is the unique conformal bijection from $H=\mathbb{H}\setminus K$ onto the complex upper half-plane with hydrodynamic normalization at infinity:The nonnegative coefficient $a_K$ is the half-plane capacity of $K$. The leading coefficient one and constant term zero remove the affine ambiguity of a half-plane map. In the compact H-hull convention, $K$ is bounded and relatively closed in $\mathbb{H}$, and its complement is a simply connected domain; boundary attachment points can be included in its closure.

We will use these standard properties explicitly: $g_K$ and its inverse extend holomorphically by Schwarz reflection principle across real intervals outside the hull's real attachment region; their real boundary restrictions are strictly increasing; and their normalized Laurent series hold in a full reflected neighbourhood of infinity. In particular, when $K$ lies in the unit disc, these extensions exist across $(-\infty ,-1)$ and $(1,\infty )$.

Write $\mathbb{H}=\{z:\mathrm{Im}z>0\}$. A compact H-hull is a bounded, relatively closed set $A\subseteq \mathbb{H}$ for which $D_A=\mathbb{H}\setminus A$ is a simply connected domain. “Compact” refers to its compact closure in $\overline{\mathbb{H}}$; boundary points on $\mathbb{R}$ can be adjoined without changing the remaining domain.

The Riemann mapping theorem and normalization at infinity give a unique mapping-out function of a compact H-hull $g_A:D_A\to \mathbb{H}$ with hydrodynamic normalization at infinity. By the Schwarz reflection principle outside a large disc, it has the Laurent seriesThe half-plane capacity is the coefficientThe empty compact H-hull has $g_A(z)=z$ and half-plane capacity zero. Nonnegativity follows from the Brownian representation of half-plane capacity proved next.

Let ${\psi }_A=g_A-g_A(0)$, and take another admissible compact H-hull $B$ in the image complex upper half-plane. Form the hull $C=A\cup {\psi }_A^{-1}(B)$, including its relative closure and filling bounded complementary components if needed. Its mapping-out function of a compact H-hull isIndeed, this composition maps the correct remaining domain onto $\mathbb{H}$, and its constant term at infinity is zero. Since ${\psi }_A(0)=0$, the chain rule givesBoth derivatives are positive by local Schwarz reflection principle and boundary orientation. Therefore the assumed avoidance formula givesThe conditioning event has positive probability because $g_A^{\prime }(0)>0$.

It remains to justify that these equalities determine the law. For a proper simple curve from $0$ to infinity, its complement has two sides, each connected to an interval of the real boundary. Any point off the curve lies in a thin polygonal tube attached to that boundary which avoids both the curve and zero; its closure, with an end cap, is an admissible compact H-hull. A countable collection of tubes with rational geometry therefore reconstructs the complement as the union of the tubes avoided by the curve.

The joint distribution of their avoidance indicators is also determined. Avoiding finitely many tubes is equivalent to avoiding the filled union when that union is admissible. If the union separates $0$ from infinity, avoidance is impossible and the joint probability is zero. Otherwise a simple curve cannot enter the bounded complementary pockets without crossing the union, so filling does not change the avoidance event. Finite joint avoidance probabilities consequently come from the same single-hull formulas; inclusion-exclusion principle supplies every finite indicator pattern. These patterns determine the random complement and hence the unparameterized simple trace.

Thus avoidance probabilities determine a simple chordal curve law, and the conditional image law above equals the original law. This proves the derivative avoidance criterion for chordal restriction, giving the chordal restriction property. Simplicity is essential to this reconstruction argument; it should not be replaced by a claim that arbitrary random closed sets are determined by these hull-avoidance tests.

Interpret the printed $A_0=0$ as the empty initial hull. Write $a(t)=\mathrm{hcap}({\tilde{A}}_t)$ and $F_t={\psi }_t$. Conformal maps preserve inclusion and simple connectedness of the complementary domains, so the image sets form an increasing hull family. Boundedness of $\psi$ on bounded sets ensures that these image hulls are bounded. The Loewner local growth property is preserved under conformal transport: after mapping out the hull at time $t$, the small new hull is transported by $F_t$ near its single boundary growth point. The Schwarz reflection principle extends $F_t$ analytically across that point, with real positive derivative. The image diameters therefore tend to zero as the original ones do. These statements use the usual hull closures and prime ends; the initial image hull is empty.

For completeness, the infinitesimal capacity rule underlying this argument is that a shrinking hull $J$ attached near $u$, transported by a map $F$ analytic there, hasOne obtains this by rescaling at $u$: the transported map tends to its linear part, and half-plane capacity scales by the square of the dilation. Uniform analytic distortion near the growth point controls the error. Applying it to the mapped-out increments gives local absolute continuity of $a$ and $a^{\prime }(t)=2F_t^{\prime }(U_t{)}^2$.

The coefficient can also be read directly from the Chordal Loewner equation. The image driver is ${\tilde{U}}_t=F_t(U_t)$. Differentiate $F_t={\tilde{g}}_t\circ \psi \circ g_t^{-1}$, at a fixed point $w$, to obtainThe left side is regular at $w=U_t$. On the right, the coefficient of $(w-U_t{)}^{-1}$ is $a^{\prime }(t)/F_t^{\prime }(U_t)-2F_t^{\prime }(U_t)$, so it must vanish. This proves the conformal change of half-plane capacity rule and its integrated version:If the derivative at the initial boundary point is singular, the formula is interpreted by integrating from a positive time and taking the lower limit to zero; finite image capacity and kernel continuity give this limit. The duplicated incomplete normalization sentence in the TeX is absent from the PDF.

A Brownian excursion in the upper half-plane from $0$ to infinity is killed planar Brownian motion conditioned to stay in $\mathbb{H}$, with its entrance law at $0$. Precisely, it is the Doob h-transform for $h(z)=\mathrm{Im}z$, obtained by starting at $i\epsilon$ and letting $\epsilon \downarrow 0$. Equivalently,where $W$ is standard Brownian motion and the independent process $R$ is a dimension-three Bessel process started at zero. The transformed generator is $\frac{1}{2}\Delta +y^{-1}{\partial }_y$, which explains this representation.

First compute avoidance from an interior point $z$. Let $\tau$ be the exit time of ordinary planar Brownian motion from $\mathbb{H}\setminus A$. The harmonic correctionhas boundary values $\mathrm{Im}z$ on $A$ and zero on the real axis. The hydrodynamic normalization at infinity gives $g_A(z)=z+\mathrm{hcap}(A)/z+O(|z{|}^{-2})$. Together with the maximum principle for harmonic functions this gives the useful bound $0\le v\le {\sup }_{a\in A}\mathrm{Im}a$. Since ordinary Brownian motion hits the real axis almost surely, the optional stopping theorem applied to this bounded harmonic function givesThe Doob h-transform weights a stopped path by its final height divided by its initial height. Applying this at the first hit of $A$, with bounded-time localization and then a limit, therefore yieldsFor the entrance law the same result follows by conditioning at a small positive time and letting that time decrease to zero: the avoidance function has the boundary limit computed next, and the excursion is initially in a neighbourhood disjoint from $A$.

Because the closure of the hull avoids $0$, the Schwarz reflection principle makes $g_A$ analytic in a neighbourhood of zero, with real $g_A(0)$ and positive real $g_A^{\prime }(0)$. ThusIn fact, the reflected Taylor expansion gives $\mathrm{Im}{g}_A(z)/\mathrm{Im}z=g_A^{\prime }(0)+O(|z|)$ uniformly as $z\to 0$ inside the half-plane, so the same limit applies to the entrance process. The map in the question is ${\psi }_A=g_A-g_A(0)$, so its derivative agrees with $g_A^{\prime }$. We obtain the restriction probability of a Brownian half-plane excursion: