Solution
= Solution
The parameterization in part (c) gives the exact self-similarity
$$
K_{\lambda^2t}=\lambda K_t
$$
for every $\lambda>0$. The <Loewner local growth property> supplies a continuous <Loewner driving function> $U$. Under this scaling of the hulls, the deterministic scaling rule for the <Chordal Loewner equation> gives
$$
U_{\lambda^2t}=\lambda U_t.
$$
Taking $t=1$ and $\lambda=\sqrt t$ shows that
$$
U_t=U_1\sqrt t=c\sqrt t
$$
for the real constant $c=U_1$.