Abstract
We consider the height process of a Lévy process with no negative jumps, and its associated continuous tree representation. Using Lévy snake tools developed by Le Gall-Le Jan and Duquesne-Le Gall, with an underlying Poisson process, we construct a fragmentation process, which in the stable case corresponds to the self-similar fragmentation described by Miermont. For the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment. We also give a special Markov property for the snake which is of its own interest.
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Abraham, R., Delmas, JF. Fragmentation associated with Lévy processes using snake. Probab. Theory Relat. Fields 141, 113–154 (2008). https://doi.org/10.1007/s00440-007-0081-2
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DOI: https://doi.org/10.1007/s00440-007-0081-2