Duality of Chordal SLE

We derive some geometric properties of chordal SLE$(\kappa ;\vec{\rho })$ 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$(\kappa ;\vec{\rho })$ process has the same distribution as the image of a chordal SLE$({\kappa }^{\mathrm{\prime }};\overrightarrow{{\rho }^{\mathrm{\prime }}})$ trace, where $\kappa >4$, ${\kappa }^{\mathrm{\prime }}=16\mathrm{/}\kappa$, and the forces $\vec{\rho }$ and $\overrightarrow{{\rho }^{\mathrm{\prime }}}$ are suitably chosen. We find that for $\kappa \ge 8$, the boundary of a standard chordal SLE$(\kappa )$ hull stopped on swallowing a fixed $x\in \mathbb{R}\text{sem}\{0\}$ is the image of some SLE$(16\mathrm{/}\kappa ;\vec{\rho })$ trace started from $x$. Then we obtain a new proof of the fact that chordal SLE$(\kappa )$ trace is not reversible for $\kappa >8$. We also prove that the reversal of SLE$(4;\vec{\rho })$ trace has the same distribution as the time-change of some SLE$(4;\overrightarrow{{\rho }^{\mathrm{\prime }}})$ trace for certain values of $\vec{\rho }$ and $\overrightarrow{{\rho }^{\mathrm{\prime }}}$.