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 $(\mathbb H,0,\infty)$.

Solution

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

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 gives
$$
\psi_D=\psi_C\circ\psi_A,
\qquad
\psi_D'(0)=\psi_C'(0)\psi_A'(0).
$$
Therefore
$$
\begin{aligned}
\mathbb P(\psi_A(\gamma)\cap C=\varnothing\mid\gamma\cap A=\varnothing)
&=\frac{\mathbb P(\gamma\cap D=\varnothing)}
{\mathbb P(\gamma\cap A=\varnothing)}\\
&=\frac{\psi_D'(0)^\alpha}{\psi_A'(0)^\alpha}
=\psi_C'(0)^\alpha.
\end{aligned}
$$
These 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>.

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

A random curve $\gamma$ from $0$ to infinity in the <complex upper half-plane> has the chordal restriction property when, for every admissible compact H-hull $A$ avoiding zero, the conditional law of $\psi_A(\gamma)$ given $\gamma\cap A=\varnothing$ equals the original law of $\gamma$. Here $\psi_A:\mathbb H\setminus A\to\mathbb H$ fixes zero and has derivative one at infinity.

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