Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 36 / 4 / a

Regard a chord as an oriented curve modulo increasing reparameterization, and write $[\gamma ]$ for that class. Scale invariance of $\mu$ means that $[r\gamma ]$ has the same law for each $r>0$. The locality of a chord law under conformal changes of neighborhoods is the following stopped-curve property: if $N,N^{\ast }$ are neighborhoods of the starting boundary point and $\Phi :N\to N^{\ast }$ is a boundary-preserving conformal isomorphism with $\Phi (0)=0$, thenThe two stopped curves are sampled from the same original law $\mu$; equality is for their unparameterized initial segments. A normalization of ${\Phi }^{\prime }(0)$ can be imposed in an equivalent formulation, since scale invariance removes the positive dilation. In a hull-removal formulation, the Locality property of SLE says that a modification away from the starting point leaves the initial law unchanged until the curve reaches the modified part of the domain.