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Superprocesses with interaction and immigration

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  • Xiong, Jie
  • Yang, Xu

Abstract

We construct a class of superprocesses with interactive branching, immigration mechanisms, and spatial motion. It arises as the limit of a sequence of interacting branching particle systems with immigration, which generalizes a result of Méléard and Roelly (1993) established for a superprocess with interactive spatial motion. The uniqueness in law of the superprocess is established under certain conditions using the pathwise uniqueness of an SPDE satisfied by its corresponding distribution function process. This generalizes the recent work of Mytnik and Xiong (2015), where the result for a super-Brownian motion with interactive immigration mechanisms was obtained. An extended Yamada–Watanabe argument is used in the proving of pathwise uniqueness.

Suggested Citation

  • Xiong, Jie & Yang, Xu, 2016. "Superprocesses with interaction and immigration," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3377-3401.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:11:p:3377-3401
    DOI: 10.1016/j.spa.2016.04.032
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    References listed on IDEAS

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    1. He, Hui, 2009. "Discontinuous superprocesses with dependent spatial motion," Stochastic Processes and their Applications, Elsevier, vol. 119(1), pages 130-166, January.
    2. Li, Zeng-Hu, 1992. "Measure-valued branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 43(2), pages 249-264, December.
    3. He, Hui & Li, Zenghu & Yang, Xu, 2014. "Stochastic equations of super-Lévy processes with general branching mechanism," Stochastic Processes and their Applications, Elsevier, vol. 124(4), pages 1519-1565.
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    Cited by:

    1. Ji, Lina & Xiong, Jie & Yang, Xu, 2023. "Well-posedness of the martingale problem for super-Brownian motion with interactive branching," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 288-322.

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