The covariant measure of SLE on the boundary


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.

Probab. Theory Related Fields
Tom Alberts
Tom Alberts
Associate Professor of Mathematics
University of Utah
Scott Sheffield
Scott Sheffield
Professor of Mathematics