Continuity of the SLE trace in simply connected domains

TL;DR

This paper proves the continuity of the SLE_ trace in simply connected domains for <8 and establishes an SLE analog of Makarov's Theorem regarding harmonic measure support.

Contribution

It demonstrates the continuity of SLE_ traces in any simply connected domain for <8 and extends Makarov's Theorem to the SLE context.

Findings

SLE_ trace is continuous in simply connected domains for <8

Established an SLE analog of Makarov's Theorem

Provides insights into harmonic measure support in SLE

Abstract

We prove that the $SLE_\kappa$ trace in any simply connected domain $G$ is continuous (except possibly near its endpoints) if $\kappa<8$. We also prove an SLE analog of Makarov's Theorem about the support of harmonic measure.