= Solution
For <Schramm–Loewner evolution> in $(\mathbb H,0,\infty)$, the <Scaling invariance of SLE> states that, for every $a>0$,
$$
\widetilde K_t=a^{-1}K_{a^2t}
$$
has the same law as $K_t$. The scaled <Loewner driving function> is $\widetilde U_t=a^{-1}U_{a^2t}$. Since $U_t=\sqrt\kappa B_t$, the <Brownian scaling> identity $a^{-1}B_{a^2t}\overset d=B_t$ 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 $\operatorname{SLE}_\kappa$ in $(\mathbb H,0,\infty)$. More precisely,
$$
\widetilde K_s=(g_t(K_{t+s}\setminus K_t)-U_t)^{\mathrm{fill}}
$$
has driving function $\widetilde U_s=U_{t+s}-U_t$. The <stationary increments> and <independent increments> of <Brownian motion> show that $\widetilde U$ 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.
Back to article page
