= Solution
The <Conformal Markov property of SLE> states that, conditional on the hull $K_t$, the image under $g_t-U_t$ of the future hull has the same law as the original hull and is independent of the past.
For a <Loewner chain> with continuous driver $U$, this property says that $U_{t+s}-U_t$ is independent of the past and has the same distribution as $U_s$. Thus $U$ has <stationary increments> and <independent increments>. Every continuous process with those properties is a Brownian motion with drift, so $U_t=at+\sigma B_t$. Conformal scale invariance gives
$$
(r^{-1}U_{r^2t})_{t\geq0}\overset d=(U_t)_{t\geq0},
$$
which forces $a=0$. Writing $\kappa=\sigma^2$ yields
$$
\boxed{U_t=\sqrt\kappa B_t}.
$$
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