The covariant measure of SLE on the boundary

Abstract

We construct a natural measure $\mu$ supported on the intersection of a chordal SLE$(\kappa)$ curve $\gamma$ with $\mathbb{R}$, in the range $4 < \kappa < 8$. The measure is a function of the SLE path in question. Assuming that boundary measures transform in a “$d$-dimensional” way (where $d$ is the Hausdorff dimension of $\gamma \cap \mathbb{R}$), we show that the measure we construct is (up to multiplicative constant) the unique measure-valued function of the SLE path that satisfies the Domain Markov property.