Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 4

The law satisfies the chordal restriction property when, for every $A\in {\mathcal{Q}}_{\pm }$, conditional on $\gamma \cap A=\varnothing$, the mapped curve ${\psi }_A(\gamma )$ has the same unparameterized law as $\gamma$ in $(\mathbb{H},0,\infty )$.

Assume the avoidance formula. Given another admissible hull $C$, put $D=A\cup {\psi }_A^{-1}(C)$ with the bounded filling. Uniqueness of the normalized maps givesThereforeThese avoidance events determine the law of a simple closed random set. They agree with those of $\gamma$, so the conditional mapped law equals the original law. Hence the avoidance formula implies the chordal restriction property.

Let $J_t={\psi }_t^{\prime }(U_t)$, $q_t={\psi }_t^{\prime \prime }(U_t)/{\psi }_t^{\prime }(U_t)$, and $r_t=({\partial }_z^3{\psi }_t)(U_t)/{\psi }_t^{\prime }(U_t)$. Since $d{U}_t=\sqrt{8/3}d{B}_t$, the supplied identity and Itô formula giveFor $M_t=J_t^{\alpha }$, its drift coefficient isThus the nonzero choice isand $M_{t\wedge \tau }$ is a continuous local martingale. The boundary Schwarz lemma for mapping-out maps gives $0\le J_{t\wedge \tau }\le 1$, so $0\le M_{t\wedge \tau }\le 1$. A bounded local martingale is a true martingale. This is the SLE eight-thirds restriction martingale.

Let $A=\mathbb{H}\cap B(1,ϵ)$. Its normalized mapping-out map isso ${\psi }_A(0)=0$ andUsing the restriction exponent $5/8$ gives

For the vertical slit $A=(1,1+iϵ]$, choose the square-root branch asymptotic to $z-1$ at infinity. The normalized map isfor which ${\psi }_A(0)=0$ andTherefore