Williams decomposition for superprocesses

We decompose the genealogy of a general superprocess with spatially dependent branching mechanism with respect to the last individual alive (Williams decomposition). This is a generalization of the main result of Delmas and H'{e}nard [Electron. J. Probab.,18,1-43,2013] where only superprocesses with spatially dependent quadratic branching mechanism were considered. As an application of the Williams decomposition, we prove that, for some superprocesses, the normalized total mass will converge to a point mass at its extinction time. This generalizes a result of Tribe [Ann. Probab.,20,286-311,1992] in the sense that our branching mechanism is more general.