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 $c=U_1$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 203 / 1 / d / Solution

{c}

The local growth property says that, after mapping out the hull at time $t$, the new hull grown during a short interval has diameter tending uniformly to zero with the interval length. It ensures that the growth is described by one continuous boundary point.


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