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$(\kappa';\vec{ \rho'})$ trace, where $\kappa>4$, $\kappa'=16/\kappa$, and the forces $\vec{ \rho}$ and $\vec{ \rho'}$ are suitably chosen. We find that for $\kappa\ge 8$, the boundary of a standard chordal SLE$(\kappa)$ hull stopped on swallowing a fixed $x\in\R\sem\{ 0\}$ is the image of some SLE$(16/\kappa;\vec{ \rho})$ 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$(4;\vec{ \rho'})$ trace for certain values of $\vec{ \rho}$ and $\vec{ \rho'}$.
Categories: math.PR