Some Properties of Annulus SLE

Abstract

An annulus SLE$_κ$ trace tends to a single point on the target circle, and the density function of the end point satisfies some differential equation. Some martingales or local martingales are found for annulus SLE$_4$, SLE$_8$ and SLE$_{8/3}$. From the local martingale for annulus SLE$_4$ we find a candidate of discrete lattice model that may have annulus SLE$_4$ as its scaling limit. The local martingale for annulus SLE$_{8/3}$ is similar to those for chordal and radial SLE$_{8/3}$. But it seems that annulus SLE$_{8/3}$ does not satisfy the restriction property.