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 the real line 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 intersected with 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.