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Catalytic Branching Processes via Spine Techniques and Renewal Theory

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2078)

Abstract

In this article we contribute to the moment analysis of branching processes in catalytic media. The many-to-few lemma based on the spine technique is used to derive a system of (discrete space) partial differential equations for the number of particles in a variation of constants formulation. The long-time behaviour is then deduced from renewal theorems and induction.

Keywords

  • Finite Variance
  • Moment Analysis
  • Renewal Theory
  • Offspring Distribution
  • Renewal Theorem

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Fig. 1

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Acknowledgements

Leif Döring would like to thank Martin Kolb for drawing his attention to [1] and Andreas Kyprianou for his invitation to the Bath-Paris workshop on branching processes, where he learnt of the many-to-few lemma from Matthew I. Roberts. The authors would also like to thank Piotr Milos for checking an earlier draft, and a referee for pointing out several relevant articles.Leif Döring acknowledges the support of the Fondation Sciences Mathématiques de Paris. Matthew I. Roberts thanks ANR MADCOF (grant ANR-08-BLAN-0220-01) and WIAS for their support.

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Döring, L., Roberts, M.I. (2013). Catalytic Branching Processes via Spine Techniques and Renewal Theory. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_12

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