WIAS Preprint No. 821, (2003)

A super-stable motion with infinite mean branching


Authors

  • Fleischmann, Klaus
  • Sturm, Anja

2010 Mathematics Subject Classification

  • 60J80 60K35 60G57 60F15

Keywords

  • Neveu's continuous state branching process, superprocess, branching processwith infinite mean, non-Lipschitz non-linearity, immortal process, instantaneous mass propagation, locally countably infinite biodiversity

DOI

10.20347/WIAS.PREPRINT.821

Abstract

Impressed by Neveu's (1992) continuous-state branching process we learned about from Bertoin and Le Gall (2000), a class of finite measure-valued càdlàg superprocesses X with Neveu's branching mechanism is constructed. To this end, we start from certain supercritical (α,d,β)-superprocesses X(β) with symmetric α-stable motion and (1+β)-branching and let β↓0. The log-Laplace equation related to X has the locally non-Lipschitz function 𝑢 log 𝑢 as non-linear term (instead of 𝑢1+β in the case of X(β)) and is thus interesting in its own. X has infinite expectations, is immortal in all finite times, propagates mass instantaneously everywhere in space, and has locally countably infinite biodiversity.

Appeared in

  • Ann. Inst. H. Poincare Probab. Statist., 40 (2004), pp. 513--537

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