Reversed radial SLE and the Brownian loop measure

TL;DR

This paper introduces a finite, conformally invariant measure related to Brownian loops intersecting radial SLE curves, and derives a Radon-Nikodym derivative formula for reversed radial SLE.

Contribution

It defines a normalized, finite measure for loops hitting radial SLE curves and establishes a Radon-Nikodym derivative formula for reversed radial SLE.

Findings

Defined a finite, conformally invariant loop measure related to radial SLE.

Derived a Radon-Nikodym derivative formula for reversed radial SLE.

Provided estimates for the measure when the curve is small in a simply connected domain.

Abstract

The Brownian loop measure is a conformally invariant measure on loops in the plane that arises when studying the Schramm-Loewner evolution (SLE). When an SLE curve in a domain evolves from an interior point, it is natural to consider the loops that hit the curve and leave the domain, but their measure is infinite. We show that there is a related normalized quantity that is finite and invariant under M\"obius transformations of the plane. We estimate this quantity when the curve is small and the domain simply connected. We then use this estimate to prove a formula for the Radon-Nikodym derivative of reversed radial SLE with respect to whole-plane SLE.