Boundary-intersection threshold for SLE ($\kappa >4$)

Chordal ${\mathrm{SLE}}_{\kappa }$ intersects the domain boundary away from its endpoints exactly when $\kappa >4$. The centered image of a boundary point is a Bessel process of dimension $1+4/\kappa$, which hits zero exactly below dimension two.

For $\kappa >4$, the Boundary-point Bessel flow for SLE and the SLE boundary swallowing criterion imply that the full Loewner trace hits the positive real axis almost surely. Every fixed open interval in that axis has positive hitting probability: its countably many dilates cover the axis, while Scaling invariance of SLE makes all their hitting probabilities equal. Reflection gives the corresponding negative-axis statement. Together with the domain Markov property of a chordal Loewner chain, this produces visits to the old Loewner trace boundary in a mapped future domain.