Biggins' Martingale Convergence for Branching Lévy Processes - Archive ouverte HAL
Article Dans Une Revue Electronic Communications in Probability Année : 2018

Biggins' Martingale Convergence for Branching Lévy Processes

Résumé

A branching Lévy process can be seen as the continuous-time version of a branching random walk; see [BM17]. It describes a particle system on the real line in which particles move and reproduce independently one of the others, in a Poissonian manner. Just as for Lévy processes, the law of a branching Lévy process is determined by its characteristic triplet (σ 2 , a, Λ), where the Lévy measure Λ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins' theorem [Big77] in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic triplet (σ 2 , a, Λ) for additive martingales of branching Lévy processes to have a non-degenerate limit. The proof is adapted from Lyons [Lyo97].
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Dates et versions

hal-01689648 , version 1 (22-01-2018)
hal-01689648 , version 2 (18-05-2019)

Identifiants

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Jean Bertoin, Bastien Mallein. Biggins' Martingale Convergence for Branching Lévy Processes. Electronic Communications in Probability, 2018, 23, ⟨10.1214/18-ECP185⟩. ⟨hal-01689648v2⟩
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