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Asymptotic behavior of solutions to the fragmentation equation with shattering: an approach via self-similar Markov processes

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Date
2010
Link to item file
http://hal.archives-ouvertes.fr/hal-00341882/en/
Dewey
Probabilités et mathématiques appliquées
Sujet
Quasi-stationary solutions; Regular Variation; Scaling Limits; Fragmentation Equation; Self-similar Markov Processes
Journal issue
The Annals of Applied Probability
Volume
20
Number
2
Publication date
2010
Article pages
382-429
Publisher
Institute of Mathematical Statistics
DOI
http://dx.doi.org/10.1214/09-AAP622
URI
https://basepub.dauphine.fr/handle/123456789/3137
Collections
  • CEREMADE : Publications
Metadata
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Author
Haas, Bénédicte
Type
Article accepté pour publication ou publié
Abstract (EN)
The subject of this paper is a fragmentation equation with non-conservative solutions, some mass being lost to a dust of zero-mass particles as a consequence of an intensive splitting. Under some assumptions of regular variation on the fragmentation rate, we describe the large-time behavior of solutions. Our approach is based on probabilistic tools: the solutions to the fragmentation equation are constructed via non-increasing self-similar Markov processes that reach continuously 0 in finite time. Our main probabilistic result describes the asymptotic behavior of these processes conditioned on non-extinction and is then used for the solutions to the fragmentation equation. We notice that two parameters influence significantly these large-time behaviors: the rate of formation of ``nearly-1 relative masses" (this rate is related to the behavior near $0$ of the L\'evy measure associated to the corresponding self-similar Markov process) and the distribution of large initial particles. Correctly rescaled, the solutions then converge to a non-trivial limit which is related to the quasi-stationary solutions to the equation. Besides, these quasi-stationary solutions, or equivalently the quasi-stationary distributions of the self-similar Markov processes, are entirely described.

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