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The probabilities that a random chordal filling avoids each admissible compact H-hull away from zero determine its law as an unparameterized closed set. Each point of its complement can be joined to the relevant real boundary arc through the complement and lies in a thin rational polygonal tube attached to that arc. These countably many hull tests recover the complement. Their joint avoidance events are tested by filled finite unions; if a union separates zero from infinity, the joint avoidance event is empty. Otherwise connectedness makes avoidance of the union equivalent to avoidance of its filled hull. Thus individual hull probabilities give every finite joint indicator distribution and determine the random complement.

For random proper simple curves from $0$ to infinity in the complex upper half-plane, the probabilities of avoiding every compact H-hull whose closure excludes zero determine the unparameterized trace law. Every point off a simple chordal curve can be connected to a real boundary interval away from zero within one side of its complement. A countable collection of thin rational polygonal tubes attached to that interval therefore recovers the complement from its avoidance indicators. Joint avoidance of finitely many tubes reduces to avoidance of their filled union. If the union blocks every path from zero to infinity the joint event is empty; otherwise filling pockets does not change the avoidance event. The inclusion-exclusion principle gives all finite indicator distributions. These determine the random complement and hence the trace. Simplicity and properness are hypotheses; the assertion is not a general determination theorem for arbitrary random closed sets.

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 random proper simple curve from zero to infinity has the displayed avoidance probability for every admissible compact H-hull $A$, with fixed $\alpha >0$. Put ${\psi }_A=g_A-g_A(0)$. For another admissible hull $B$ in the image complex upper half-plane, map composition and the chain rule give $g_C^{\prime }(0)=g_B^{\prime }(0)g_A^{\prime }(0)$ for $C=A\cup {\psi }_A^{-1}(B)$. The conditional image curve therefore has avoidance probabilities $g_B^{\prime }(0{)}^{\alpha }$, exactly those of the original. Since avoidance probabilities determine a simple chordal curve law, the conditional image law agrees with the original. This is the chordal restriction property.

If a compact H-hull lies in a disc of radius $r$ centred at $\xi \in \mathbb{R}$, the displayed estimate holds for $|z-\xi |>2r$, with an absolute constant $C$. It refines the Laurent expansion uniformly in the hull and supplies the first-order increment needed to derive the Chordal Loewner equation from the Loewner local growth property.

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

For $x\in \mathbb{R}$ and $R>0$, the filled half-disc $K=\{z\in \mathbb{H}:|z-x|\le R\}$ is a compact H-hull. Its mapping-out function of a compact H-hull is $g_K(z)=z+R^2/(z-x)$. On the semicircle the image is real, and the inverse branch asymptotic to the identity maps the complex upper half-plane onto the exterior half-disc. Its Laurent series has coefficient $R^2/z$, so $\boxed{\mathrm{hcap}(K)=R^2}$. Using an open half-disc instead requires taking its relative closure first.

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.

For a compact H-hull and a Borel subset of the intrinsic boundary of a simply connected domain, hydrodynamic normalization at infinity and the Poisson kernel for the upper half-plane give this limit as $y\to \infty$ with $x/y\to 0$. After multiplying the kernel by $\pi y$, it tends pointwise to one and is eventually uniformly bounded. Dominated convergence proves the finite-measure case and Fatou's lemma proves the infinite-measure case.

For a compact H-hull, $v(z)=\mathrm{Im}{g}_K(z)$ is harmonic and satisfies $0<v(z)\le \mathrm{Im}z$. Local boundedness and the interior continuity of $g_K^{-1}$ show that $v$ tends to zero at every finite boundary point. The maximum principle for harmonic functions applied to $v-\mathrm{Im}z$, with an exhaustion controlling infinity through the Laurent expansion, proves the inequality without boundary smoothness.

An increasing family of compact H-hulls of half-plane capacity $2t$ with the Loewner local growth property corresponds uniquely to a continuous real Loewner driving function. Its mapping-out functions solve the Chordal Loewner equation up to the maximal lifetime of each point. Conversely each continuous driver generates that hull family. A continuous Loewner trace is an additional property, not a consequence for every continuous driver. For SLE it is supplied by the trace-existence theorem.

For the compact H-hull $(x,x+ih]$, the square-root branch asymptotic to $z-x$ gives the hydrodynamic normalization at infinity of its mapping-out function of a compact H-hull. Its expansion has coefficient $h^2/2$, the half-plane capacity of a vertical slit.

For nested compact H-hulls whose closures avoid $x\in \mathbb{R}$, the boundary derivative is an excursion avoidance probability. Avoiding the larger hull is a subset of avoiding the smaller one, which proves the derivative inequality. For a hull contained in $\{|z|\le R\}$ and $|x|>R$, comparison with the filled half-disc gives the sharp bound $1-R^2/x^2\le g_A^{\prime }(x)\le 1$, since that half-disc has mapping-out function $z+R^2/z$. The empty hull and filled half-disc attain the two bounds.

With $m(z)=(z+1)/(1-z)$, the compact H-hull $K_L=m^{-1}(i(0,L])$ is a unit semicircular arc attached at $-1$, with a narrow passage near $1$. As $L\to \infty$, its normalized mapping-out function of a compact H-hull tends to $2$ at each fixed point of the interior bay. Points in that bay tending to $-1$ therefore have displacement approaching three. The order of these two limits is important.

A compact H-hull is a bounded relatively closed subset $K$ of $\mathbb{H}$ with simply connected complement, considered together with its compact closure in $\overline{\mathbb{H}}$. Its unique mapping-out function has hydrodynamic normalization at infinityThe nonnegative coefficient $a_K$ is its half-plane capacity. Thus the half-plane-capacity parameterization here gives $g_t(z)=z+2t/z+O(z^{-2})$. Strict increase means $K_s⊊K_t$ whenever $s<t$; the strictly increasing capacity already excludes equality.

The Loewner local growth property means that the new growth becomes small after removing the old compact H-hull. More precisely set $K_{t,t+h}=\overline{g_t(K_{t+h}\setminus K_t)}$, taking the closure from the complex upper half-plane. On each bounded time interval, the diameters of these image compact H-hulls tend uniformly to zero as $h\downarrow 0$. Equivalently the new growth can be separated from infinity by crosscuts of vanishing diameter in the mapped domain. It is this single-point local growth, together with capacity continuity, that gives a continuous real Loewner driver.

The Loewner transform ${\xi }_t$ is the limiting point at which this mapped growth occurs:The limit is independent of the chosen new-growth points. The Loewner correspondence theorem says that the resulting mapping-out functions satisfyConversely, given a continuous real $\xi$, solve this ordinary differential equation until the maximal lifetime $T_z$ of each $z\in \mathbb{H}$. The surviving set $H_t=\{z:T_z>t\}$ is mapped conformally onto $\mathbb{H}$ by $g_t$; its complement, with the appropriate boundary closure, is $K_t$. The Laurent coefficient is $2t$, and the composition law reconstructs the local growth. A continuous Loewner driver always determines the compact H-hulls; generation by a continuous Loewner trace is an additional property, not a conclusion for every continuous Loewner driver.

A chordal SLE Loewner trace in the complex upper half-plane with parameter $\kappa \ge 0$ is the continuous curve associated with the Loewner driver ${\xi }_t=\sqrt{\kappa }{B}_t$, where $B$ is standard real Brownian motion started at zero. It starts at zero, its compact H-hull at time $t$ is the curve's past with all pockets cut off from infinity filled, andThe capacity clock has $\mathrm{hcap}(K_t)=2t$. For $\kappa =0$ this gives the vertical slit $\gamma (t)=2i\sqrt{t}$.

To prove Scaling invariance of SLE, fix $r>0$ and putThe Laurent series of ${\hat{g}}_t$ has coefficient $2t$, and direct differentiation givesBrownian scaling says $(r^{-1}{B}_{r^{2}t}{)}_{t\ge 0}$ has the same law as $B$. The Loewner drivers therefore agree in law, and uniqueness of the Chordal Loewner equation gives equality in law of the maps, compact H-hulls, and continuous Loewner traces:This identifies the time change $t\mapsto r^{2}t$ as well as the spatial scaling.

For a simply connected Jordan domain $D$ with distinct marked boundary points, choose a conformal bijection $f:\mathbb{H}\to D$ taking zero to the initial point and infinity to the target, and use $f(\gamma )$ as an unparameterized curve. The Caratheodory boundary extension theorem supplies the boundary values of $f$. Any second such map is $f\circ (z\mapsto rz)$ for $r>0$, because the automorphisms of $\mathbb{H}$ fixing zero and infinity are positive dilations. The Scaling invariance of SLE just proved makes the two image laws identical, up to the corresponding deterministic change of capacity clock. Thus this defines SLE consistently in marked Jordan domains. The assumed $|\gamma (t)|\to \infty$ and the continuous boundary extension give $f(\gamma (t))\to f(\infty )$, the specified target. The law is conformally natural and does not impose a distinguished half-plane-capacity parameterization on an arbitrary domain.

Let ${\xi }_t=\sqrt{\kappa }{B}_t$ and $0<\kappa <4$. We use these precise Bessel process facts, which the question permits: for dimension $\delta >2$ a process started at $r>0$ exists for all time, never reaches zero, and has a strictly positive all-time minimum. More quantitatively,No assertion about SLE simplicity or transience is being assumed.

For a real point $x>0$ before its Loewner lifetime, set $Z_t^x=g_t(x)-{\xi }_t$. Its equation isThus $Z^x/\sqrt{\kappa }$ is a Bessel process of dimension $1+4/\kappa >2$. For each rational $x>0$ it stays positive on every finite interval, almost surely. These countably many events hold together. They cover every real $y>0$ as well: choose rational $0<x<y$ and observe that the noise cancels in the difference,Consequently $Z^y$ cannot reach zero while $Z^x$ remains positive. The negative side has the identical argument with the reflected Brownian motion.

This excludes Loewner trace contact with every real point other than zero, rather than just with each fixed rational point. On any finite interval a positive gap at a real point stays bounded away from zero. Continuous dependence of the Chordal Loewner equation then solves it on a complex neighborhood of that point, and the analytic flow extends the map conformally across a real interval there. Its real derivative is $\exp (-2{\int }_0^t(Z_s^x{)}^{-2}ds)>0$, and its image stays away from the driving point. The Loewner trace tip therefore cannot be there. We have proved $\gamma [0,\infty )\cap \mathbb{R}\subseteq \{0\}$.

Now use Brownian motion independent increments to restart at every deterministic rational time $s>0$. The centered future maps have Loewner driver ${\xi }_{s+t}-{\xi }_s$ and therefore the original SLE law, independently of the past. Its Loewner trace stays in $\mathbb{H}\cup \{0\}$ by the preceding argument. The inverse map $g_s^{-1}$ extends continuously to the real boundary for a continuous trace-generated compact H-hull: the finite Loewner trace gives a locally connected boundary, and the Caratheodory boundary extension theorem applies. Its value at ${\xi }_s$ is $\gamma (s)$. Thus the future in the original plane can meet $K_s\cup \mathbb{R}$ only at $\gamma (s)$. All these restart conclusions hold simultaneously for rational $s$.

Suppose $\gamma (u)=\gamma (v)$ with $u<v$. The Loewner trace cannot be constant throughout $(u,v)$, since its compact H-hull capacity increases strictly. By continuity there is a rational $s\in (u,v)$ with $\gamma (s)\ne \gamma (u)$. But $\gamma (v)=\gamma (u)$ lies in $K_s\cup \mathbb{R}$, contradicting the restart conclusion at $s$. This also excludes any return to the starting point. Hence the Loewner trace is simple and lies in the open complex upper half-plane at every positive time.

For transience, first strengthen the boundary conclusion to avoidance of the closure at each fixed real point. Fix $x>0$ and $0<r<x/4$. If the now-simple curve first enters the closed half-disc of radius $r$ about $x$ at time $\sigma$, join its tip to $x$ by the segment inside that disc. Before $\sigma$ the segment misses the Loewner trace. Together with the Loewner trace and the real interval from zero to $x$, it bounds a pocket whose boundary includes the positive side of the Loewner trace and $[0,x/2]$. A Brownian motion path from high on the imaginary axis reaching that boundary arc must enter the pocket through the segment. The maximum principle for harmonic functions bounds its harmonic measure by the chance of hitting the entire half-disc.

The boundary arc maps under $g_{\sigma }$ to $[{\xi }_{\sigma },g_{\sigma }(x/2)]$. Its harmonic measure from $iY$, as $Y\to \infty$, isThe half-disc maps out by $z\mapsto z+r^2/(z-x)$; its semicircle becomes an interval of length $4r$. Its harmonic measure has asymptotic $4r/(\pi Y)$. Comparing and letting $Y\to \infty$ yieldsBut $m_x={\inf }_{t\ge 0}(g_t(x/2)-{\xi }_t)>0$ by the all-time minimum of a transient Bessel process fact. The curve cannot enter any such disc with $4r<m_x$. Therefore it stays a strictly positive distance from $x$ for all time. Reflection proves the same for fixed $x<0$. This is the boundary approach forces a small SLE Bessel gap argument; the crosscut comparison is also described in the boundary-closure lemma in the primary SLE paper.

Map out the simple initial slit at time one. The starting point zero has two distinct real prime end images $b_{-},b_{+}$ under $g_1$, lying on opposite sides of ${\xi }_1$. Conditionally on the initial slit, the image future is a fresh independent SLE translated by ${\xi }_1$. The just-proved fixed-point closure avoidance applies to each of $b_{-}-{\xi }_1$ and $b_{+}-{\xi }_1$. Therefore neither $b_{-}$ nor $b_{+}$ lies in the closure of the mapped future. If a sequence of future Loewner trace points approached zero in the original domain, their images would approach one of these two prime ends. This is impossible. ThusFinally Scaling invariance of SLE gives $R_T:={\inf }_{t\ge T}|\gamma (t)|\stackrel{d}{=}\sqrt{T}{R}_1$. For every fixed $R>0$,The events of returning to the closed radius-$R$ disc after time $T$ decrease as $T$ increases. Continuity of probability on decreasing events implies that the probability of returns at arbitrarily late times is zero. Taking the countable intersection over positive integer $R$ provesThis proves actual eventual escape, not merely unboundedness of the growing compact H-hulls.

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.

For chordal SLE with parameter $\kappa$, the Loewner trace $\gamma$ starts at zero and is parametrized by half-plane capacity $2t$. Let $D_t$ be the unbounded component of $\mathbb{H}\setminus \gamma [0,t]$, and let $K_t=\mathbb{H}\setminus D_t$ be its filled compact H-hull. When the trace is not simple, the hull includes the regions it disconnects from infinity. The associated mapping-out functions of compact H-hulls have expansionand satisfy the Chordal Loewner equationwhere $B$ is standard real Brownian motion. The equation is solved until $z$ is swallowed by the hull. The Loewner transform $\xi$ records the real boundary point at which the mapped hull grows; its relation to the trace isEquivalently the tip maps to ${\xi }_t$ in the appropriate boundary prime-end sense. Thus the flow encodes the surviving domains, the transform drives that flow, and the trace generates its hulls. For $\kappa =0$, the driver is identically zero, $g_t(z)=\sqrt{z^2+4t}$ with the branch asymptotic to $z$, and ${\gamma }_t=2i\sqrt{t}$.

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.