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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-203/1/d/solution