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Path processes and historical superprocesses
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  • Published: March 1991

Path processes and historical superprocesses

  • E. B. Dynkin1 

Probability Theory and Related Fields volume 90, pages 1–36 (1991)Cite this article

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Summary

A superprocessX over a Markov process ξ can be obtained by a passage to the limit from a branching particle system for which ξ describes the motion of individual particles.The historical process \(\hat \xi \) for ξ is the process whose state at timet is the path of ξ over time interval [0,t]. The superprocess\(\hat X\) over\(\hat \xi \) the historical superprocess over ξ—reflects not only the particle distribution at any fixed time but also the structure of family trees. The principal property of a historical process\(\hat \xi \) is that\(\hat \xi _s \) is a function of\(\hat \xi _t \) for alls<t. Every process with this property is calleda path process. We develop a theory of superprocesses over path processes whose core is the integration with respect to measure-functionals. By applying this theory to historical superprocesses we construct the first hitting distributions and prove a special Markov property for superprocesses.

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Authors and Affiliations

  1. Department of Mathematics, White Hall, Cornell University, 14853-7901, Ithaca, NY, USA

    E. B. Dynkin

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  1. E. B. Dynkin
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Partially supported by National Science Foundation Grant DMS-8802667

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Dynkin, E.B. Path processes and historical superprocesses. Probab. Th. Rel. Fields 90, 1–36 (1991). https://doi.org/10.1007/BF01321132

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  • Received: 15 March 1990

  • Issue Date: March 1991

  • DOI: https://doi.org/10.1007/BF01321132

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AMS 1980 subject classification

  • Primary 60J80
  • 60G57
  • secondary 60J25
  • 60J50
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