Loewner differential equation

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.