Existence and Uniqueness of Nonsimple Multiple SLE

We prove the existence and uniqueness of multiple \(\hbox {SLE}_{\kappa }\) associated with any given link pattern for \(\kappa \in (4,6]\). We also have the uniqueness for \(\kappa \in (6,8)\). The multiple \(\hbox {SLE}_{\kappa }\) law is constructed by first inductively constructing a \(\sigma \)-finite multiple \(\hbox {SLE}_{\kappa }\) measure and then normalizing the measure whenever it is finite. The total mass of the measure satisfies the conformal covariance, asymptotics and PDE for multiple \(\hbox {SLE}_{\kappa }\) partition functions in the literature subject to the assumption that it is smooth.