= Solution
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 infinity>
$$
g_K:\mathbb H\setminus K\longrightarrow\mathbb H,
\qquad g_K(z)=z+\frac{a_K}{z}+O(z^{-2})\quad(z\to\infty).
$$
The 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 $K_s\subsetneq K_t$ 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 $h\downarrow0$. 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:
$$
\boxed{\xi_t=\lim_{h\downarrow0}g_t(z_h),\qquad z_h\in(K_{t+h}\setminus K_t)\cap\mathbb H.}
$$
The limit is independent of the chosen new-growth points. The <Loewner correspondence theorem> says that the resulting <mapping-out functions> satisfy
$$
\partial_tg_t(z)=\frac{2}{g_t(z)-\xi_t},\qquad g_0(z)=z.
$$
Conversely, 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 $\kappa\geq0$ 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, and
$$
\gamma(t)=\lim_{y\downarrow0}g_t^{-1}(\xi_t+iy).
$$
The capacity clock has $\operatorname{hcap}(K_t)=2t$. For $\kappa=0$ this gives the vertical slit $\gamma(t)=2i\sqrt t$.
To prove <Scaling invariance of SLE>, fix $r>0$ and put
$$
\widehat K_t=r^{-1}K_{r^2t},\qquad
\widehat g_t(z)=r^{-1}g_{r^2t}(rz),\qquad
\widehat\xi_t=r^{-1}\xi_{r^2t}.
$$
The <Laurent series> of $\widehat g_t$ has coefficient $2t$, and direct differentiation gives
$$
\partial_t\widehat g_t(z)=\frac2{\widehat g_t(z)-\widehat\xi_t}.
$$
<Brownian scaling> says $(r^{-1}B_{r^2t})_{t\geq0}$ 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>:
$$
\boxed{(r^{-1}\gamma(r^2t))_{t\geq0}\overset d=(\gamma(t))_{t\geq0}.}
$$
This identifies the time change $t\mapsto r^2t$ 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.
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