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Champs de Lévy additifs spectralement positifs et processus de branchement multi-types continus

Abstract : The discrete time and space branching processes (BP) have been introduced at the end of the XIXth century in order to study the extinction of a population. In the 1960’s, J. Lamperi and M. Jiřina generalise this BP to a continuous time and space setting (CBP). In 1967, J. Lamperti shows that the CBP can be represented with a spectrally positive Lévy process (spLp). The asymptotic behaviour of the CBP is characterized by the Laplace exponent of this spLp, called branching mechanism. This branching mechanism enables us to state the Grey’s condition which distinguishes the finite and infinite extinction of the CBP. In this thesis, we are interested in the multitypes CBP (MCBP) i.e. an individual bears a type which characterises its behaviour. A.E. Kyprianou, S. Palau, M. Barczy, G. Pap have studied the asymptotics of MCBP. Nevertheless, there was no multi-type Grey’s condition. In other words, we could not distinguish finite and infinite extinction in multi-type case. The aim of this works was to establish a multi-type Grey’s condition. To this end, we usedthe Lamperti representation which represents the CMBP with a particular Lévy field. In a first hand, we had to study the fluctuation theory of these fields in order to use its hitting time of negative levels. Finally, we established a multi-type Grey’s condition for a MCBP which characterize its extinction in terms of the ones of its diagonal CBP.
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Marine Marolleau. Champs de Lévy additifs spectralement positifs et processus de branchement multi-types continus. Mathématiques générales [math.GM]. Université d'Angers, 2020. Français. ⟨NNT : 2020ANGE0074⟩. ⟨tel-03637078⟩

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