Loewner chain

A chordal Loewner chain is an increasing family of compact H-hulls, usually parameterized by half-plane capacity, whose mapping-out functions evolve according to the Chordal Loewner equation.

Conditionally on the entire past up to $t$, the future domains mapped and centered by $g_t-U_t$ have the original chain law and are independent of that past. For Schramm–Loewner evolution this follows from independent stationary Brownian motion increments and the composition rule for chordal Loewner driving functions. The Strong Markov property gives the version at almost surely finite stopping times.

Suppose a chordal SLE trace stays in the open complex upper half-plane at every positive time almost surely. For every deterministic rational time $s$, the domain Markov property of a chordal Loewner chain makes its mapped and centered future a fresh SLE, so its positive-time future stays in the open half-plane as well. The inverse mapping-out function then places that future outside the entire old hull. Intersecting these probability-one events over rational $s$ rules out every self-intersection: if two times $u<v$ shared a trace point, choose rational $s$ strictly between them. This proves simplicity using the allowed boundary-avoidance assertion for $0<\kappa \le 4$.

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.

When the indicated limit exists continuously and the hulls are generated by its past, it defines the trace of the Loewner chain. The hull $K_t$ includes the curve up to time $t$ and any regions it disconnects from infinity. Therefore swallowing a point by $K_t$ is different from visiting that point by the Loewner trace. For SLE, the continuous trace is part of its basic existence theory.

For a continuous Loewner trace generating its hulls, the inverse mapping-out function of a compact H-hull $g_t^{-1}$ extends continuously to the closed complex upper half-plane. The finite trace is a continuous map image of an interval, and the boundary of its unbounded complementary component has local connectedness; the Caratheodory boundary extension theorem gives the extension. In particular $g_t^{-1}(U_t)=\gamma (t)$, while $g_t^{-1}$ maps the open complex upper half-plane into $D_t$. This is the deterministic boundary-extension input of Rohde and Schramm, Theorem 4.1. Under a Conformal Markov property of SLE restart, this relates boundary contacts of the mapped future to contacts with the old hull, without assuming the old trace is simple.

The local growth property says that, after mapping out the hull at time $t$, the new hull grown during a short interval has diameter tending uniformly to zero with the interval length. It ensures that the growth is described by one continuous boundary point.

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.

The Loewner differential equation describes a growing family of simply connected planar domains through ordinary differential equations for their normalized conformal maps.

For a capacity-parameterized locally growing hull family,where the continuous real function $U$ is the Loewner driving function.

With half-plane capacity $2t$, $f_t=g_t-W_t$ and $G_{\mathbb{H}}(z,w)=\log |(z-\overline{w})/(z-w)|$, differentiating $G_{D_t}(z,w)=G_{\mathbb{H}}(f_t(z),f_t(w))$ gives the displayed identity. The translation by the driver cancels in the Green function. This rank-one covariance loss balances the quadratic covariation of the harmonic mean in the SLE4 coupling with a Gaussian free field.

For a real point $b$ away from the initial driver, this is the maximal lifetime of the real solution ${\partial }_t{g}_t(b)=2/(g_t(b)-{\xi }_t)$. At finite lifetime its centred image collides with the Loewner driving function. A point may be visited or separated from infinity by the hull, so swallowing is not the same as an individual trace visit.

For a continuous chordal SLE trace started at zero, hitting the real interval $[b,\infty )$ is equivalent to finite boundary-point swallowing time for a Loewner chain at $b>0$. A boundary crosscut may swallow all points between its endpoints without visiting them individually. The Boundary-point Bessel flow for SLE determines the probability of this event.

For a continuous Loewner driving function $U$, $T_z$ is the maximal lifetime of $g_t(z)$ in the complex upper half-plane under the Chordal Loewner equation. A finite lifetime has $\mathrm{Im}{g}_t(z)\to 0$ as $t\uparrow T_z$: otherwise the continuous driver and the ordinary differential equation continue the solution. A swallowed point need not have been visited by the Loewner trace; it may lie in a disconnected pocket of the filled hull.

The Loewner driving function $U_t$ is the real boundary point at which the mapped-out hull grows. Scaling the hulls by $a^{-1}$ and time by $a^2$ transforms it to $a^{-1}{U}_{a^{2}t}$.

Suppose the mapped increments $K_{s,t}$ have real-centred enclosing radii bounded by a nondecreasing modulus $\omega (t-s)\to 0$. Nested closures define the unique growing real point ${\xi }_s$. For $v\in K_{t,t+h}$, its preimage $z=g_{K_{s,t}}^{-1}(v)$ belongs to $K_{s,t+h}$. The uniform displacement estimate for a mapping-out function gives $|v-z|\le C\omega (t-s)$, while $|z-{\xi }_s|\le 2\omega (t-s+h)$. Letting $h\downarrow 0$ with $h\le t-s$ proves the displayed estimate. It proves uniform continuity of the Loewner transform on each bounded time interval without evaluating a conformal map on a rough boundary.

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.

After time $t$, map remaining domains by $g_t-U_t$. The new mapping-out function at elapsed time $s$ is ${\hat{g}}_s(z)=g_{t+s}(g_t^{-1}(z+U_t))-U_t$. Direct differentiation of the Chordal Loewner equation proves that its driver is $U_{t+s}-U_t$, and its half-plane capacity is $2s$. Defining transformed hulls as complements of transformed domains avoids ambiguity at the old hull boundary.

For the Chordal Loewner equation driven by $U_t=a\sqrt{t}$, the hull is a straight slit whose endpoint is proportional to $\sqrt{t}$. Put $r_{\pm }=(a\pm \sqrt{a^2+16})/2$ and $\alpha =-r_{-}/(r_{+}-r_{-})\in (0,1)$. Its inverse map isusing upper-half-plane branches. The interval between the two real roots maps onto the two sides of a segment of argument $\pi \alpha$.

For a real $x$ before its swallowing time,It lies in $(0,1]$ and decreases with time.