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Feller Property and Infinitesimal Generator of the Exploration Process

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Abstract

We consider the exploration process associated to the continuous random tree (CRT) built using a Lévy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale.

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References

  1. Abraham, R., Delmas, J.-F.: Fragmentation associated with Lévy processes using snake. Preprint CERMICS (2005)

  2. Aldous, D.: The continuum random tree III. Ann. Probab. 21(1), 248–289 (1993)

    MATH  MathSciNet  Google Scholar 

  3. Bertoin, J.: Lévy processes. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  4. Duquesne, T., Le Gall, J.-F.: Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002)

  5. Ethier, S.N., Kurtz, T.G.: Markov Processes. Wiley, New York (1986)

    MATH  Google Scholar 

  6. Le Gall, J.-F., Le Jan, Y.: Branching processes in Lévy processes: Laplace functionals of snake and superprocesses. Ann. Probab. 26, 1407–1432 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Le Gall, J.-F., Le Jan, Y.: Branching processes in Lévy processes: the exploration process. Ann. Probab. 26, 213–252 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 2nd edn. Springer, New York (1995)

    Google Scholar 

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Authors and Affiliations

  1. MAPMO, Université d’Orléans, B.P. 6759, 45067, Orleans cedex 2, France

    Romain Abraham

  2. ENPC-CERMICS, 6-8 av. Blaise Pascal, Champs-sur-Marne, 77455, Marne La Vallee, France

    Jean-François Delmas

Authors
  1. Romain Abraham
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  2. Jean-François Delmas
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Corresponding author

Correspondence to Romain Abraham.

Additional information

The research of the second author was partially supported by NSERC Discovery Grants of the Probability group at Univ. of British Columbia.

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Abraham, R., Delmas, JF. Feller Property and Infinitesimal Generator of the Exploration Process. J Theor Probab 20, 355–370 (2007). https://doi.org/10.1007/s10959-007-0082-1

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  • DOI: https://doi.org/10.1007/s10959-007-0082-1

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