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

At time zero,

Υ_{0} (z) = Im z

. Compactness of

K \subset H

gives

ϵ_{0} = min_{z \in K} Im z > 0.

Also

M_{0} (z)

is uniformly bounded above on

K

. On

E_{ϵ} = {τ_{ϵ} < \infty, S_{τ_{ϵ}} \geq 1/2},

one has

M_{τ_{ϵ}} \geq ϵ^{- p} 2^{- q}

. Optional stopping, Fatou lemma, and the assumed conditional angular estimate give

sup_{z \in K} M_{0} (z) \geq E [M_{τ_{ϵ}}; E_{ϵ}] \geq ϵ^{- p} 2^{- q} c_{1} P (τ_{ϵ} < \infty) .

Thus

P_{z} (τ_{ϵ} < \infty) \leq C_{K} ϵ^{(8 - κ) /8} .

The exponent

(8 - κ) / κ

requested in the question does not follow and is false as written. The SLE Green-function estimate gives probability comparable to

ϵ^{1 - κ /8}

, confirming that the denominator in the requested exponent should be

8

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