A Williams decomposition for spatially dependent superprocesses. (English) Zbl 1294.60104

Summary: We present a genealogy for superprocesses with a non-homogeneous quadratic branching mechanism, relying on a weighted version of the superprocess introduced by J. Engländer and R. G. Pinsky [Ann. Probab. 27, No. 2, 684–730 (1999; Zbl 0979.60078)] 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 superprocess from the root, and define another process by looking from the top. Examples including the multitype Feller diffusion (investigated by N. Champagnat and S. Roelly [Electron. J. Probab. 13, 777–810 (2008; Zbl 1189.60154)]) and the superdiffusion are provided.


60J68 Superprocesses
60J25 Continuous-time Markov processes on general state spaces
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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