Reversed Radial SLE and the Brownian Loop Measure

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 Möbius 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.