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Survival of homogeneous fragmentation processes with killing. (English. French summary) Zbl 1301.60087

The authors consider a homogeneous fragmentation process in which there is an additional killing upon crossing a certain space-time barrier (the killing mechanism is defined in Section 1.2).
After an introduction, in Section 2 the authors provide some general notions that are used in the subsequent part of the paper and they discuss the connection between fragmentation and Lévy processes.
Then, Section 3 is concerned with the proof of Theorem 2 and in Section 4 they provide the proof of Theorem 3, presented in the Introduction. To this end, the authors use an auxiliary lemma which states that, for any \(n\in \mathbb N\), there exists a time such that with positive probability the fragmentation process has at least \(n\) blocks.
In the next section, they introduce a multiplicative process and examine when this process is a martingale.
Then the authors consider an additive process which also turns out to be a martingale and whose limit is studied with regard to strict positivity.
Finally, the authors deal with the asymptotic behaviour of the largest fragment in the killed fragmentation process.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60G09 Exchangeability for stochastic processes
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References:

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