= Solution
Under the Loewner correspondence, mapping out the past replaces the driver by
$$
\widetilde U_s=U_{t+s}-U_t.
$$
The conformal Markov property therefore makes $U$ a continuous process with stationary independent increments. Every such process has the form
$$
U_t=at+\sqrt\kappa B_t.
$$
Scale invariance of SLE says that $r^{-1}U_{r^2t}$ has the same law as $U_t$. Comparing means forces $ar=a$ for every $r>0$, hence $a=0$; comparison of variances leaves the constant $\kappa$. Nondegeneracy gives $\kappa>0$, so
$$
U_t=\sqrt\kappa B_t.
$$
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