SLE coordinate changes

TL;DR

This paper introduces a unified framework for different types of Schramm-Loewner Evolution (SLE), showing how radial, chordal, and dipolar SLE are interconnected through extended definitions and martingale descriptions.

Contribution

It extends the definition of SLE with force points into the interior of the domain, unifying radial, chordal, and dipolar SLE under a common framework.

Findings

Radial SLE(kappa) is equivalent to chordal SLE(kappa;rho) with rho=kappa-6.

Martingales for Radon-Nikodym derivatives are explicitly described.

Unified description simplifies understanding of different SLE variants.

Abstract

The purpose of this note is to describe a framework which unifies radial, chordal and dipolar SLE. When the definition of SLE(kappa;rho) is extended to the setting where the force points can be in the interior of the domain, radial SLE(kappa) becomes chordal SLE(kappa;rho), with rho=kappa-6, and vice versa. We also write down the martingales describing the Radon-Nykodim derivative of SLE(kappa;rho_1,...,rho_n) with respect to SLE(kappa).