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The critical branching diffusion in a varying environment

Published online by Cambridge University Press:  01 July 2016

B. Gail Ivanoff
B. Gail Ivanoff*
Affiliation:
University of Ottawa
*
∗Postal address: Department of Mathematics, University of Ottawa, Ottawa, Ontario, Canada K1N 9B4.
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Abstract

A critical branching diffusion in an environment which varies in time is defined. Conditions are given for convergence of the process to a limit, and the triviality or non-triviality of the limit is discussed. If the migration is a Gaussian process in two dimensions, the limit is trivial, and an exponential limit law is proven.

Keywords

POINT PROCESSPROBABILITY GENERATING FUNCTIONALLAPLACE TRANSFORMFACTORIAL MOMENTSFACTORIAL CUMULANTS
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 3 , September 1983 , pp. 488 - 506
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

References

Athreya, K. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes: Statistical Analysis, Theory, and Applications, ed. Lewis, P. A. W., Wiley, New York, 299–383.Google Scholar
Durrett, R. (1979) An infinite particle system with additive interactions. Adv. Appl. Prob. 11, 355–383.CrossRefGoogle Scholar
Fleischman, J. (1978) Limiting distributions for branching random fields. Trans. Amer. Math. Soc. 239, 353–389.Google Scholar
Ivanoff, B. G. (1980a) The branching diffusion with immigration. J. Appl. Prob. 17, 1–15.Google Scholar
Ivanoff, B. G. (1980b) The branching random field. Adv. Appl. Prob. 12, 825–847.Google Scholar
Ivanoff, B. G. (1982) The multitype branching random walk I. Canad. J. Statist. To appear.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley-Interscience, New York.Google Scholar
Kallenberg, O. (1977) Stability of critical cluster fields. Math. Nachr. 77, 7–43.Google Scholar
Klein, B. and Macdonald, P. D. M. (1980) The multitype continuous-time branching process in a periodic environment. Adv. Appl. Prob. 12, 81–93.Google Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, New York.Google Scholar
Moyal, J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 1–31.Google Scholar
Tanabe, H. (1979) Equations of Evolution. Pitman, London.Google Scholar
Westcott, M. (1972) The probability generating functional. J. Austral. Math. Soc. 14, 448–466.CrossRefGoogle Scholar