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>.

Consequently
$$
\{\tau_x>\tau_y\}=\{\text{the SLE trace reaches }(-\infty,y]\text{ before }[x,\infty)\}.
$$
It is the event that the negative marked point is swallowed before the positive marked point.

Solution

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

{c}

For chordal $\operatorname{SLE}_\kappa$, the centered image of a real boundary point, divided by $\sqrt\kappa$, evolves as a <Bessel process> of dimension
$$
d=1+\frac4\kappa.
$$
Its first hit of zero is the time at which the point is swallowed by the Loewner hull.

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 Incoming links: Boundary-point Bessel flow for SLE