The Covariant Measure of SLE on the Boundary

TL;DR

This paper constructs a natural boundary measure supported on the intersection of chordal SLE(kappa) curves with the real line for 4<kappa<8, demonstrating its uniqueness and domain Markov property under certain assumptions.

Contribution

It introduces a canonical boundary measure for SLE curves intersecting the real line and proves its uniqueness based on the domain Markov property.

Findings

Constructed a boundary measure supported on SLE intersection with R.

Proved the measure's uniqueness under domain Markov property assumptions.

Established the measure's dependence on the SLE path.

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.