Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 203 / 3 / e / Solution

At $\kappa =4$, the Boundary-point Bessel flow for SLE has dimension two. Its scale function of a one-dimensional diffusion is $s(x)=\log x$, givingThe same bounded-path and countable-union argument as in part (c) excludes a finite zero hit. Hence the critical case belongs to the boundary-avoiding regime:The countable argument of part (d) gives no visits to any positive real point almost surely. By Reflection invariance of Brownian motion, reflected SLE has the same law, so no negative real point is visited either. Thus the set of real points visited is exactly $\{0\}$. Recurrence of a two-dimensional Bessel process near zero does not mean that it hits zero.