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Journal of Theoretical Probability

, Volume 20, Issue 2, pp 355–370 | Cite as

Feller Property and Infinitesimal Generator of the Exploration Process

  • Romain Abraham
  • Jean-François Delmas
Article

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.

Keywords

Exploration process Lévy snake Feller property Measure valued process Infinitesimal generator 

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References

  1. 1.
    Abraham, R., Delmas, J.-F.: Fragmentation associated with Lévy processes using snake. Preprint CERMICS (2005) Google Scholar
  2. 2.
    Aldous, D.: The continuum random tree III. Ann. Probab. 21(1), 248–289 (1993) MATHMathSciNetGoogle Scholar
  3. 3.
    Bertoin, J.: Lévy processes. Cambridge University Press, Cambridge (1996) MATHGoogle Scholar
  4. 4.
    Duquesne, T., Le Gall, J.-F.: Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) Google Scholar
  5. 5.
    Ethier, S.N., Kurtz, T.G.: Markov Processes. Wiley, New York (1986) MATHGoogle Scholar
  6. 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) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Le Gall, J.-F., Le Jan, Y.: Branching processes in Lévy processes: the exploration process. Ann. Probab. 26, 213–252 (1998) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 2nd edn. Springer, New York (1995) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.MAPMO, Université d’OrléansOrleans cedex 2France
  2. 2.ENPC-CERMICSMarne La ValleeFrance

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