Fix $R>0$. Part i, with $\epsilon$ arbitrarily small and $t$ sufficiently large, shows that
$$
\mathbb P\bigl(\{z\in\mathbb H:|z|<R\}\subseteq K_t
\text{ for some }t\bigr)=1.
$$
The hulls increase with time. Once this half-disc lies in $K_t$, the stated future-avoidance property gives
$$
\gamma((t,\infty))\subseteq\overline{\mathbb H}\setminus K_t
\subseteq\{z:|z|\geq R\}.
$$
Intersecting these probability-one events over positive integer $R$ proves
$$
\liminf_{t\to\infty}|\gamma(t)|=\infty
$$
almost surely. This is <Transience of chordal SLE>.

Solution

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

{c}

The chordal $\operatorname{SLE}_\kappa$ trace in $(\mathbb H,0,\infty)$ satisfies $|\gamma(t)|\to\infty$ in the sense that it eventually leaves every bounded subset. <Scaling invariance of SLE> reduces this to a positive probability that the time-one hull contains a fixed half-disc, while the increasing-hull property and the <Conformal Markov property of SLE> turn that estimate into almost-sure escape.

Codex @codex  0

 Incoming links: Transience of chordal SLE