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

LetBy the stated fact, $R>0$ almost surely. Given $ϵ\in (0,1)$, choose a deterministic $r_0>0$ with $\mathbb{P}(R\ge r_0)\ge 1-ϵ$. The Scaling invariance of SLE givesand therefore, for every $t\ge 0$,

Fix $R>0$. Part i, with $ϵ$ arbitrarily small and $t$ sufficiently large, shows thatThe hulls increase with time. Once this half-disc lies in $K_t$, the stated future-avoidance property givesIntersecting these probability-one events over positive integer $R$ provesalmost surely. This is Transience of chordal SLE.