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Let $h_t$ intertwine two Loewner chains, with driver ${\xi }_t=\sqrt{\kappa }{B}_t$, and set $a_t=h_t^{\prime }({\xi }_t)$, $b_t=h_t^{\prime \prime }({\xi }_t)$, $c_t=h_t^{\prime \prime \prime }({\xi }_t)$. Expanding the conjugacy equation at the driver gives ${\partial }_t{h}_t^{\prime }({\xi }_t)=b_t^2/(2a_t)-4c_t/3$; the Itô formula gives the displayed diffusion. For a real exponent $p$, the drift of $a_t^p$ is $p[1+\kappa (p-1)]a_t^{p-2}{b}_t^2/2+p(\kappa /2-4/3)a_t^{p-1}{c}_t$. At $\kappa =8/3$, $p=5/8$ cancels both terms and yields the SLE eight-thirds restriction martingale.

For a Loewner chain with half-plane capacity $2t$, scale invariance means $({\lambda }^{-1}{K}_{{\lambda }^{2}t}{)}_{t\ge 0}$ has the original law for every $\lambda >0$. Its mapping-out maps are ${\lambda }^{-1}{g}_{{\lambda }^{2}t}(\lambda z)$ and its Loewner driving function is ${\lambda }^{-1}{U}_{{\lambda }^{2}t}$. This follows by differentiating the Chordal Loewner equation. Brownian scaling gives this property for Schramm–Loewner evolution.

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.

For conjugated Loewner chains, ${\varphi }_t={\tilde{g}}_t\circ \varphi \circ g_t^{-1}$ and ${\tilde{\xi }}_t={\varphi }_t({\xi }_t)$. The image capacity speed is $2{\varphi }_t^{\prime }({\xi }_t{)}^2$. Differentiating this identity gives ${\dot{\varphi }}_t(z)=2{\varphi }_t^{\prime }({\xi }_t{)}^2/({\varphi }_t(z)-{\varphi }_t({\xi }_t))-2{\varphi }_t^{\prime }(z)/(z-{\xi }_t)$. The removable singularity at the driver has value $-3{\varphi }_t^{\prime \prime }({\xi }_t)$.

A crosscut of a planar domain is a simple open arc inside the domain whose two endpoints are distinct boundary points, interpreted as prime ends when necessary. In a simply connected domain it divides the domain into two components. In the half-plane, a bounded crosscut joining two real points separates the intervening real interval from infinity; a Loewner chain completing that crosscut swallows this interval.

The domains of a capacity-parametrized Loewner chain determine their hydrodynamically normalized mapping-out functions of compact H-hulls. At a surviving point, the Chordal Loewner equation gives $U_t=g_t(z)-2/{\partial }_t{g}_t(z)$. For a continuous driver and $t>0$, the left derivative uses only the hull past. High points $in$, $n\in \mathbb{N}$, survive up to any fixed finite time for sufficiently large $n$, making this a countable local reconstruction of the driver and its past filtration.

Let $t(u)$ be the inverse image-capacity clock. Part (a) and the quadratic variation of a stochastic integral giveThe Dambis-Dubins-Schwarz theorem, or the Lévy characterization of Brownian motion, now shows that ${\tilde{\xi }}_u/\sqrt{6}$ is standard Brownian motion up to the terminal image time. If that terminal clock is finite, it can be extended by a fresh Brownian continuation; this does not assert anything about the subsequent image trace. The initial value is zero because $\Phi (0)=0$.

Differentiating the image Loewner chain with respect to its own half-plane capacity time givesThus, until its first hit of $[1,\infty )$, the reparametrized image $\Phi (\gamma )$ has the ordinary chordal SLE law from $0$ to infinity. This follows from the uniqueness of the Chordal Loewner equation for a given continuous driver; the continuous Loewner trace is recovered from the inverse-map limit. Localization can be removed by exhausting times strictly before this boundary contact. The terminal event is defined from the trace itself, so equality of the stopped trace laws follows as well.

On the other hand, the definition of Conformal invariance of SLE says that the full unparameterized image $\Phi (\gamma )$ has the law of chordal ${\mathrm{SLE}}_6$ from $\Phi (0)=0$ to $\Phi (\infty )=1$, namely the law of ${\gamma }^{\prime }$. Its stopping event at $[1,\infty )$ is exactly the image of the original stopping event at $(-\infty ,-1]$. Therefore the same stopped image has both laws: ordinary ${\mathrm{SLE}}_6$ from $0$ to infinity stopped at $T$, and ${\mathrm{SLE}}_6$ from $0$ to $1$ stopped at $T^{\prime }$. We concludeThe statement does not identify $T$ with $T^{\prime }$ in their original clocks. It is target-change locality of SLE6, derived from the zero drift and capacity time change rather than assumed as a theorem.

A compact H-hull is a bounded set $K\subset \mathbb{H}$ that is closed relative to the complex upper half-plane and whose complement $D=\mathbb{H}\setminus K$ is a simply connected domain. Its Euclidean closure is compact in $\overline{\mathbb{H}}$; “compact” here does not require separation from the real axis. One may equivalently describe the closed hull in $\overline{\mathbb{H}}$, with its real-boundary convention understood. Its mapping-out function is the unique conformal map to $\mathbb{H}$ with hydrodynamic normalization at infinity.

Use the capacity convention $\mathrm{hcap}(K_t)=2t$. A chordal Schramm–Loewner evolution from $0$ to infinity is the hull family of the Chordal Loewner equationwhere $W$ is a real standard Brownian motion and $\kappa \ge 0$. The points with interior-point swallowing time for a Loewner chain at most $t$ constitute $K_t$. For each $t$, $g_t$ maps $\mathbb{H}\setminus K_t$ conformally onto $\mathbb{H}$ and has expansion $g_t(z)=z+2t/z+O(z^{-2})$. The case $\kappa =0$ is the deterministic vertical-slit chain.

First identify the relevant transformed Loewner driving functions directly. For $\lambda >0$, defineDifferentiation gives ${\partial }_t{\tilde{g}}_t=2/({\tilde{g}}_t-{\lambda }^{-1}{U}_{{\lambda }^{2}t})$. Its initial value is $z$, its domain is $\mathbb{H}\setminus {\lambda }^{-1}{K}_{{\lambda }^{2}t}$, and its expansion is $z+2t/z+O(z^{-2})$. Thus its driving function is ${\lambda }^{-1}{U}_{{\lambda }^{2}t}$, by uniqueness of the differential equation and of the hydrodynamically normalized map. Brownian scaling provesThis is capacity-parametrized scale invariance of a Loewner chain.

For a fixed time $t$, map the future remaining domains by $z\mapsto g_t(z)-U_t$. The corresponding hulls ${\hat{K}}_s$ are specified without any boundary-image ambiguity byTheir mapping-out functions areAt $s=0$ this is the identity. Differentiating at a point in its domain yieldsThe Laurent series is $z+2s/z+O(z^{-2})$. Hence the transformed Loewner driver is exactly $U_{t+s}-U_t$, with the original capacity clock unchanged. This proves the composition rule for chordal Loewner driving functions, rather than assuming an identification of transforms.

The Brownian motion increments after $t$ are independent of its past and have the original law. Since past hulls are measurable functions of the past Loewner driver, conditionally on the past, the centered mapped future is an independent copy of the original chain. This is the domain Markov property of a chordal Loewner chain. At an almost surely finite stopping time it follows likewise from the Strong Markov property.

For the converse, a precise class is essential: take capacity-parametrized Loewner chains generated by a continuous real Loewner driving function $U$, with $U_0=0$, and require the two displayed hull properties, including independence from the entire hull past in the domain Markov property. Within this class they characterize ${\mathrm{SLE}}_{\kappa }$, for some $\kappa \ge 0$.

To see why the hull assertions determine the Loewner driver assertions, the hull domains determine their normalized maps uniquely. For any surviving point,The left time derivative suffices for $t>0$, since $U$ is continuous. A sufficiently high point survives up to any given finite time, so points $in$, $n\in \mathbb{N}$, suffice to reconstruct the Loewner driver locally and measurably from the hull past. Thus the driving-function reconstruction for a chordal Loewner chain identifies the two past filtrations and makes the preceding scale and composition computations reversible.

The domain Markov property therefore gives stationary independent increments for $U$. Its continuity makes it a continuous Lévy process. Its classification gives $U_t=bt+\sqrt{\kappa }{W}_t$: in the Lévy–Khintchine formula continuity removes the jump measure, leaving characteristic function $\exp (t(ibu-\kappa u^2/2))$. Driver scaling would turn the drift $b$ into $\lambda b$, so invariance for every $\lambda$ forces $b=0$. The diffusion coefficient is unchanged. This proves the scale-and-domain-Markov characterization of SLE with its regularity and parametrization hypotheses stated explicitly.

A family $(K_t{)}_{t\ge 0}$ of compact H-hulls is non-decreasing whenThis permits equal hulls at different times. A Loewner chain with strictly increasing half-plane capacity has strict inclusion for $s<t$. Usually its initial compact H-hull is $K_0=\varnothing$.

Work with a locally growing Loewner chain started at $0$, with $A_0=\varnothing$ and the half-plane-capacity parameterization $\mathrm{hcap}(A_t)=2t$. The Conformal Markov property of SLE combines invariance under conformal changes of the marked domain with restarting after an initial hull has been removed.

In the half-plane this means, first, that $(r^{-1}{A}_{r^{2}t}{)}_{t\ge 0}$ has the same law as $(A_t{)}_{t\ge 0}$ for every $r>0$. Second, conditionally on ${\mathcal{F}}_t$, the future hulls mapped by $g_t-U_t$,with the usual hull closure, have the original law and are independent of ${\mathcal{F}}_t$. Their capacity is $2s$ by the half-plane-capacity composition rule. One may require the same restarting property at finite stopping times.

Equivalently, the driver has the scaling law and conditional increment lawThe conformal invariance clause matters: the restarting property alone would also allow a Brownian driver with deterministic drift.

The scaling rule for the Chordal Loewner equation shows that $A_t=\sqrt{t}{A}_1$, because $r^{-1}{U}_{r^{2}t}=a\sqrt{t}$. To identify the shape, rather than infer it from scaling alone, construct the inverse map explicitly. PutThen $r_{-}<0<r_{+}$, $r_{-}{r}_{+}=-4$, and $\alpha r_{+}+(1-\alpha )r_{-}=0$. Defineusing logarithms whose arguments lie in $(0,\pi )$ on $\mathbb{H}$. Its Laurent series is $f_t(w)=w-2t/w+O(w^{-2})$, so it has the inverse hydrodynamic normalization at infinity.

On the real interval $(r_{-}\sqrt{t},r_{+}\sqrt{t})$, the first factor has argument $\pi \alpha$ and the second has argument zero. The image therefore lies on one straight ray. Its modulus increases from zero to a maximum at $w=a\sqrt{t}$, and then decreases to zero. The complementary real intervals map to the negative and positive real axes. The argument principle, or the usual conformal slit-map construction, shows that $f_t$ maps $\mathbb{H}$ conformally onto the half-plane minus that segment; the interval traverses its two sides.

The boundary walk has winding number one around every point of the half-plane off the segment: the two traversals of the slit cancel, leaving the real boundary and a large semicircle. The argument principle therefore gives exactly one preimage of each such point.

It remains to check that this is the correct Loewner chain. For $f=f_1$,Together with $f_t(w)=\sqrt{t}f(w/\sqrt{t})$, this givesthe inverse form of the Chordal Loewner equation. Uniqueness therefore identifies its slit with $A_t$. Its endpoint is $\sqrt{t}{f}_1(a)$, where $f_1(a)\ne 0$ lies in $\mathbb{H}$. This proves that the square-root Loewner driving function generates a straight slit and thata straight ray from the boundary point $0$ to infinity.

The Conformal Markov property of SLE states that, conditional on the hull $K_t$, the image under $g_t-U_t$ of the future hull has the same law as the original hull and is independent of the past.

For a Loewner chain with continuous driver $U$, this property says that $U_{t+s}-U_t$ is independent of the past and has the same distribution as $U_s$. Thus $U$ has stationary increments and independent increments. Every continuous process with those properties is a Brownian motion with drift, so $U_t=at+\sigma B_t$. Conformal scale invariance giveswhich forces $a=0$. Writing $\kappa ={\sigma }^2$ yields

For a continuous real driving function $U$, solve the Chordal Loewner equationup to the first time $T_z$ at which $g_t(z)-U_t$ tends to zero. The generated hulls areThey form an increasing Loewner chain of compact H-hulls, satisfy $\mathrm{hcap}(K_t)=2t$, and $g_t$ is the mapping-out function of $K_t$.

It is enough by Scaling invariance of SLE and reflection in the imaginary axis to treat $x=1$. Take $r=1+ϵ$. The drifttends to $(\kappa -8)/2<0$ as $z\to -\infty$. Choose $L<0$ and $a<0$ such that $b(z)\le a$ for $z\le L$. Before $\tilde{Z}$ reaches $L$,The infinite-horizon crossing probability for Brownian motion with negative drift shows that this process has a finite running maximum almost surely, so

Order preservation for the Loewner flow gives ${\tau }_1\le {\tau }_{1+ϵ}$. On $\{{\tau }_1<{\tau }_{1+ϵ}\}$, one has $V_t^1\to 0$ while $V_t^{1+ϵ}$ remains positive, so ${\tilde{Z}}_t\to +\infty$ and in particular $T_L<\infty$. HenceIf the trace itself hit the fixed boundary point $1$, then $1$ would be the right endpoint of the swallowed interval and ${\tau }_1<{\tau }_{1+ϵ}$ for every $ϵ>0$. Its probability is therefore zero. Scaling and reflection prove that SLE does not hit a fixed nonzero boundary point for every $x\in \mathbb{R}\setminus \{0\}$.

For Schramm–Loewner evolution in $(\mathbb{H},0,\infty )$, the Scaling invariance of SLE states that, for every $a>0$,has the same law as $K_t$. The scaled Loewner driving function is ${\tilde{U}}_t=a^{-1}{U}_{a^{2}t}$. Since $U_t=\sqrt{\kappa }{B}_t$, the Brownian scaling identity $a^{-1}{B}_{a^{2}t}\stackrel{d}{=}{B}_t$ proves the claim.

The Conformal Markov property of SLE states that, conditionally on the hull through time $t$, the future hull mapped by $g_t-U_t$ is an independent ${\mathrm{SLE}}_{\kappa }$ in $(\mathbb{H},0,\infty )$. More precisely,has driving function ${\tilde{U}}_s=U_{t+s}-U_t$. The stationary increments and independent increments of Brownian motion show that $\tilde{U}$ is independent of ${\mathcal{F}}_t$ and has the same law as $\sqrt{\kappa }B$. The deterministic correspondence between continuous drivers and Loewner chains completes the proof.

Set $\kappa =4/(d-1)>4$. For chordal Schramm–Loewner evolution, the centered image of a real boundary point, divided by $\sqrt{\kappa }$, follows the Boundary-point Bessel flow for SLE; changing $B$ to $-B$ matches the sign convention in the question. Thus ${\tau }_x$ and ${\tau }_y$ are the times at which the marked boundary points $x$ and $y$ are swallowed, or equivalently disconnected from infinity, by the Loewner chain.

ConsequentlyIt is the event that the negative marked point is swallowed before the positive marked point.

The trace of ${\mathrm{SLE}}_{\kappa }$ is simple for $0<\kappa \le 4$, has self-intersections without being space-filling for $4<\kappa <8$, and is space-filling for $\kappa \ge 8$. The thresholds arise from the Boundary-point Bessel flow for SLE and estimates for the Loewner chain near its driving point.

A prime end describes an approach to the boundary of a simply connected domain through a nested sequence of crosscuts. A conformal map induces a correspondence between prime ends, even when the Euclidean boundary does not have a continuous one-to-one parametrization. This is the natural boundary interpretation of the endpoints and growing tip of a Loewner chain.

Among Loewner chains with capacity $2t$, continuous real driver and initial driver zero, scale invariance and the centered domain Markov property of a chordal Loewner chain characterize Schramm–Loewner evolution with some $\kappa \ge 0$. Driving-function reconstruction for a chordal Loewner chain turns the hull properties into stationary independent increments and Brownian scaling for the driver. The continuous Lévy process classification then gives $U_t=bt+\sqrt{\kappa }{W}_t$, and scaling eliminates $b$. Regularity and parametrization are essential hypotheses of this characterization.

Chordal Schramm–Loewner evolution in the complex upper half-plane from $0$ to $\infty$ is the random Loewner chain driven by $U_t=\sqrt{\kappa }{B}_t$, where $B$ is standard Brownian motion and $\kappa \ge 0$.