Abstract
We give a stochastic model for the fragmentation phase of an avalanche. We construct a fragmentation-branching process related to the avalanches, on the set of all fragmentation sizes introduced by Bertoin. A fractal property of this process is emphasized. We also establish a specific stochastic differential equation of fragmentation. It turns out that specific branching Markov processes on finite configurations of particles with sizes bigger than a strictly positive threshold are convenient for describing the continuous time evolution of the number of the resulting fragments. The results are obtained by combining analytic and probabilistic potential theoretical tools.
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Acknowledgments
The authors acknowledge enlightening discussions with Ioan R. Ionescu and Jean-Stéphane Dhersin, during the preparation of this paper. This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0045. For the third author the research was financed through the project “Excellence Research Fellowships for Young Researchers”, the 2015 Competition, founded by the Research Institute of the University of Bucharest (ICUB).
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Beznea, L., Deaconu, M. & Lupaşcu, O. Stochastic Equation of Fragmentation and Branching Processes Related to Avalanches. J Stat Phys 162, 824–841 (2016). https://doi.org/10.1007/s10955-015-1432-5
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DOI: https://doi.org/10.1007/s10955-015-1432-5
Keywords
- Fragmentation kernel
- Avalanche
- Branching process
- Stochastic differential equation of fragmentation
- Space of fragmentation sizes