= Solution
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 $\operatorname{hcap}(K_t)=2t$. A chordal <Schramm–Loewner evolution> from $0$ to infinity is the hull family of the <Chordal Loewner equation>
$$
\boxed{\partial_tg_t(z)=\frac{2}{g_t(z)-U_t},\quad g_0(z)=z,\qquad U_t=\sqrt\kappa\,W_t,}
$$
where $W$ is a real standard <Brownian motion> and $\kappa\geq0$. 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 $\lambda>0$, define
$$
\widetilde g_t(z)=\lambda^{-1}g_{\lambda^2t}(\lambda z).
$$
Differentiation gives $\partial_t\widetilde g_t=2/(\widetilde g_t-\lambda^{-1}U_{\lambda^2t})$. Its initial value is $z$, its domain is $\mathbb H\setminus\lambda^{-1}K_{\lambda^2t}$, and its expansion is $z+2t/z+O(z^{-2})$. Thus its driving function is $\lambda^{-1}U_{\lambda^2t}$, by uniqueness of the differential equation and of the hydrodynamically normalized map. <Brownian scaling> proves
$$
\boxed{(\lambda^{-1}K_{\lambda^2t})_{t\geq0}\ \overset{d}=\ (K_t)_{t\geq0}.}
$$
This 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 $\widehat K_s$ are specified without any boundary-image ambiguity by
$$
\mathbb H\setminus\widehat K_s
=g_t(\mathbb H\setminus K_{t+s})-U_t.
$$
Their <mapping-out functions> are
$$
\widehat g_s(z)=g_{t+s}\bigl(g_t^{-1}(z+U_t)\bigr)-U_t.
$$
At $s=0$ this is the identity. Differentiating at a point in its domain yields
$$
\partial_s\widehat g_s(z)
=\frac{2}{\widehat g_s(z)-(U_{t+s}-U_t)}.
$$
The <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>, \b[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 \b[$\operatorname{SLE}_\kappa$, for some $\kappa\geq0$].
To see why the hull assertions determine the <Loewner driver> assertions, the hull domains determine their normalized maps uniquely. For any surviving point,
$$
U_t=g_t(z)-\frac{2}{\partial_tg_t(z)}.
$$
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.
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