Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 2 / c

Putso $M_t={{\rm Y}}_t^{-p}{S}_t^q$. Before ${\tau }_{ϵ}$, one has ${{\rm Y}}_t\ge ϵ$ and $0<S_t\le 1$. ThereforeThe supplied continuous local martingale is thus bounded after stopping, and a bounded local martingale is a true martingale. Hence

At time zero, ${{\rm Y}}_0(z)=\mathrm{Im}z$. Compactness of $K\subset \mathbb{H}$ givesAlso $M_0(z)$ is uniformly bounded above on $K$. Onone has $M_{{\tau }_{ϵ}}\ge {ϵ}^{-p}{2}^{-q}$. Optional stopping, Fatou lemma, and the assumed conditional angular estimate giveThusThe exponent $(8-\kappa )/\kappa$ requested in the question does not follow and is false as written. The SLE Green-function estimate gives probability comparable to ${ϵ}^{1-\kappa /8}$, confirming that the denominator in the requested exponent should be $8$.

For compact $K\subset \mathbb{H}$, letBy the Tonelli theorem and part (ii),Every point in the SLE range has conformal radius tending to zero and therefore belongs to every $E_{ϵ}$. Hence the range inside $K$ has zero expected Lebesgue measure, and so has zero measure almost surely. Exhausting $\mathbb{H}$ by countably many compact sets proves