A Williams decomposition for spatially dependent super-processes

We present a genealogy for super-processes with a non-homogeneous quadratic branching mechanism, relying on a weighted version of the super-process introduced by Engländer and Pinsky and a Girsanov theorem. We then decompose this genealogy with respect to the last individual alive (Williams' decomposition). Letting the extinction time tend to infinity, we get the Q-process by looking at the super-process from the root, and define another process by looking from the top. Examples including the multitype Feller diffusion (investigated by Champagnat and Roelly) and the super-diffusion are provided.