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

The Phase classification of the SLE trace is:

For a real boundary point $x\ne 0$, the Boundary-point Bessel flow for SLE says that $V_t^x/\sqrt{\kappa }$ is a Bessel process of dimensionWhen $\kappa \le 4$, one has $d\ge 2$, so part 2(b), including its logarithmic $d=2$ case, shows that no fixed boundary point is swallowed.

If the trace touched the real line away from its starting point, or if a later segment crossed an earlier segment, the resulting hull would disconnect from infinity a nonempty real interval, which contains a rational number. Applying the preceding argument after every rational time and using the Conformal Markov property of SLE rules this out on a countable probability-one event. Continuity of the trace then shows that no two distinct times have the same image. Thus ${\mathrm{SLE}}_{\kappa }$ is simple for $0<\kappa \le 4$.

Let ${\hat{B}}_t=r{B}_{t/r^2}$. By Brownian scaling, $\hat{B}$ is standard Brownian motion. Substituting $Y_t=r{X}_{t/r^2}$ into the Bessel equation givesWeak uniqueness for the Bessel equation shows that $Y$ is a Bessel process of dimension $d$ started at $rx$. This is the Scaling invariance of a Bessel process.

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.