The covariant measure of SLE on the boundary

Abstract

We construct a natural measure μ supported on the intersection of a chordal SLE(κ) curve γ with R, in the range 4 < κ < 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 γ ∩ 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. © 2009 Springer-Verlag.