Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 203 / 3 / a / Solution

Set $\kappa =4/(d-1)>4$. For chordal Schramm–Loewner evolution, the centered image of a real boundary point, divided by $\sqrt{\kappa }$, follows the Boundary-point Bessel flow for SLE; changing $B$ to $-B$ matches the sign convention in the question. Thus ${\tau }_x$ and ${\tau }_y$ are the times at which the marked boundary points $x$ and $y$ are swallowed, or equivalently disconnected from infinity, by the Loewner chain.

ConsequentlyIt is the event that the negative marked point is swallowed before the positive marked point.