Regularity Properties of a Supercritical Superprocess

  • S. E. Kuznetsov
Part of the Progress in Probability book series (PRPR, volume 34)

Abstract

The objective of this chapter is to derive the regularity properties of a superprocess from those of the spatial process. We assume that the spatial process ζ is a right (nonhomogeneous) Markov process in a Luzin state space and we prove that a superprocess over ζ can be chosen right if the first moments of the total mass X t (E) are finite.

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References

  1. [DM]
    C. Dellacherie and P.-A. Meyer, Probabilités et Potentiel, Hermann, Paris, 1975, 1980, 1983, 1987.MATHGoogle Scholar
  2. [Dyl]
    E.B. Dynkin, The initial and final behavior of trajectories of Markov processes, Russian Math. Surveys 26,4 (1971), 165–185; Reprinted in “London Mathematical Society Lecture Note Series 54,” 123–143, Cambridge Univ. Press, London, New York.MathSciNetCrossRefGoogle Scholar
  3. [Dy2]
    E.B. Dynkin, Integral representation of excessive measures and excessive functions, Russian Math. Surveys 27,1 (1972), 43–84; Reprinted in “London Mathematical Society Lecture Note Series 54,” 145–186, Cambridge Univ. Press, London, New York.MathSciNetCrossRefMATHGoogle Scholar
  4. [Dy3]
    E.B. Dynkin, Regular Markov processes, Russian Math. Surveys 28,2 (1973), 33–64; Reprinted in “London Mathematical Society Lecture Note Series 54,” 187–218, Cambridge Univ. Press, London, New York.MathSciNetCrossRefMATHGoogle Scholar
  5. [Dy4]
    E.B. Dynkin, Sufficient statistics and extreme points, Annals of Probability 6 (1978), 705–730.MathSciNetCrossRefMATHGoogle Scholar
  6. [Dy5]
    E.B. Dynkin, Regular transition functions and regular superprocesses, Trans. Amer. Math. Society 316 (1989), 623–634.MathSciNetCrossRefMATHGoogle Scholar
  7. [Dy6]
    E.B. Dynkin, Birth delay of a Markov process and the stopping distributions for regular processes, Probab. Theory Relat. Fields 94 (1993), 399–411.MathSciNetCrossRefMATHGoogle Scholar
  8. [Dy7]
    E.B. Dynkin, On regularity of superprocesses, Probab. Theory Relat. Fields 95 (1993), 263–281.MathSciNetCrossRefMATHGoogle Scholar
  9. [Dy8]
    E.B. Dynkin, An Introduction to Branching Measure-Valued Processes, Centre de Recherches Mathématiques of Université de Montreal and American Mathematical Society, 1994 (forthcoming book).Google Scholar
  10. [F]
    P.J. Fitzsimmons, Construction and regularity of measure-valued Markov branching processes, Israel J. Math. 64 (1988), 337–361.MathSciNetCrossRefMATHGoogle Scholar
  11. [K]
    K. Kuratowski, Topology, Academic Press, New York, 1967.Google Scholar
  12. [Ku]
    S.E. Kuznetsov, Nonhomogeneous Markov processes, Sovremennye Problemy Matematiki 20 (1982), 37–178, VINITI, Moscow (Russian); English translation in J. Soviet Math. 25 (1984), 1380–1498.Google Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • S. E. Kuznetsov
    • 1
  1. 1.Central Institute for Economics and Mathematics of The Russian Academy of SciencesMoscowRussia

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