Duality of Chordal SLE, II

TL;DR

This paper enhances understanding of the duality properties of SLE processes, especially for <, by analyzing boundary behaviors and limits of SLE traces, revealing new geometric insights.

Contribution

It provides improved geometric properties of SLE(;\u2192) processes and characterizes the boundary of SLE(,) hulls at stopping times, advancing the duality theory.

Findings

Boundary of SLE() hulls is an SLE() trace image.

Limit of SLE(;) traces exists for many cases.

Describes boundary behavior of SLE hulls for <.

Abstract

We improve the geometric properties of SLE$(\kappa;\vec{\rho})$ processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for $\kappa\in (4,8)$, the boundary of a standard chordal SLE$(\kappa)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/\kappa;\vec{\rho})$ trace started from a random point. Using this fact together with a similar proposition in the case that $\kappa\ge 8$, we obtain a description of the boundary of a standard chordal SLE$(\kappa)$ hull for $\kappa>4$, at a finite stopping time. Finally, we prove that for $\kappa>4$, in many cases, the limit of a chordal or strip SLE$(\kappa;\vec{\rho})$ trace exists.