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Extinction times of multitype, continuous-state branching processes

Abstract : A multitype continuous-state branching process (MCSBP) Z = (Z t) t≥0 , is a Markov process with values in [0, ∞) d that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of d Laplace exponents of R d-valued spectrally positive Lévy processes, each one having d − 1 increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing on the branching mechanism holds. This condition extends Grey's condition that is well known for d = 1. Our arguments bear on elements of fluctuation theory for spectrally positive additive Lévy fields recently obtained in [7] and an extension of the Lamperti representation in higher dimension proved in [5].
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https://hal.archives-ouvertes.fr/hal-03336113
Contributor : Loïc Chaumont Connect in order to contact the contributor
Submitted on : Monday, September 6, 2021 - 10:19:54 PM
Last modification on : Wednesday, October 20, 2021 - 3:18:54 AM
Long-term archiving on: : Tuesday, December 7, 2021 - 7:31:49 PM

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Loïc Chaumont, Marine Marolleau. Extinction times of multitype, continuous-state branching processes. 2021. ⟨hal-03336113⟩

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