Let
$$
R=\sup\{r>0:\{z\in\mathbb H:|z|<r\}\subseteq K_1\}.
$$
By the stated fact, $R>0$ almost surely. Given $\epsilon\in(0,1)$, choose a deterministic $r_0>0$ with $\mathbb P(R\geq r_0)\geq1-\epsilon$. The <Scaling invariance of SLE> gives
$$
K_t\overset d=\sqrt t\,K_1,
$$
and therefore, for every $t\geq0$,
$$
\mathbb P\!\left(
\{z\in\mathbb H:|z|<r_0\sqrt t\}\subseteq K_t
\right)\geq1-\epsilon.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 3 / d / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-203/3/d/i/solution