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The complete characterization of a general class of superprocesses
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  • Published: March 2000

The complete characterization of a general class of superprocesses

  • Guillaume Leduc1 

Probability Theory and Related Fields volume 116, pages 317–358 (2000)Cite this article

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Abstract.

In an important paper, [7], Dynkin, Kuznetsov and Skorohod showed that, under mild conditions, the log-Laplace functional of every branching measure-valued process is the solution of an evolution equation determined by three parameters, ξQ and ℓ. This paper essentially deals with the converse of this result. We consider a general class ℋ of BMV (subject to some mild conditions). First, we derive from [7] that every process X∈ℋ is a (ξΦ, K)-superprocesses, where the triples (ξ, Φ, K) are subject to some conditions ANS1-ANS3. Conversely, we show that, for each of these triples satisfying ANS1-ANS3, there exists a version X of the (ξ, Φ, K)-superprocess which belongs to ℋ. Consequently, ANS1-ANS3 fully characterizes ℋ, and subject to mild conditions, BMV and superprocesses are equivalent concepts. This requires to prove a general existence theorem for superprocesses, the existence of a regular version of these processes and that for processes in ℋ, branching characteristics Q and ℓ are continuous.

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Authors and Affiliations

  1. Université du Québec à Montréal, Département de Mathématiques, Case Postale 8888, Succursale Centre-Ville, Montréal (Québec), Canada H3C 3P8. e-mail: leduc@math.uqam.ca, , , , , , CA

    Guillaume Leduc

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  1. Guillaume Leduc
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Received: 11 April 1996 / Revised version: 25 May 1999

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Leduc, G. The complete characterization of a general class of superprocesses. Probab Theory Relat Fields 116, 317–358 (2000). https://doi.org/10.1007/s004400050252

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  • Issue Date: March 2000

  • DOI: https://doi.org/10.1007/s004400050252

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Keywords

  • Evolution Equation
  • General Class
  • Mild Condition
  • Existence Theorem
  • Complete Characterization
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