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

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

2008-11-26

arxiv.org/abs/0811.4267v2

Annals of Applied Probability 2010, Vol. 20, No. 2, 382-429

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AAP622 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/09-AAP622

The subject of this paper is a fragmentation equation with nonconservative 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 nonincreasing self-similar Markov processes that continuously reach 0 in finite time. Our main probabilistic result describes the asymptotic behavior of these processes conditioned on nonextinction and is then used for the solutions to the fragmentation equation. We note that two parameters significantly influence 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 with the corresponding self-similar Markov process) and the distribution of large initial particles. Correctly rescaled, the solutions then converge to a nontrivial limit which is related to the quasi-stationary solutions of the equation. Besides, these quasi-stationary solutions, or, equivalently, the quasi-stationary distributions of the self-similar Markov processes, are fully described.

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