Duality of Chordal SLE

Abstract

We derive some geometric properties of chordal SLE$(κ;\vecρ)$ processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE$(κ;\vecρ)$ process has the same distribution as the image of a chordal SLE$(κ';\vec{ρ'})$ trace, where $κ>4$, $κ'=16/κ$, and the forces $\vecρ$ and $\vec{ρ'}$ are suitably chosen. We find that for $κ\ge 8$, the boundary of a standard chordal SLE$(κ)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/κ;\vecρ)$ trace started from $x$. Then we obtain a new proof of the fact that chordal SLE$(κ)$ trace is not reversible for $κ>8$. We also prove that the reversal of SLE$(4;\vecρ)$ trace has the same distribution as the time-change of some SLE$(4;\vec{ρ'})$ trace for certain values of $\vecρ$ and $\vec{ρ'}$.