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

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 $0<\kappa \le 4$.