A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process - ENS - École normale supérieure Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2022

A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process

Quan Shi

Résumé

A continuous-time particle system on the real line verifying the branching property and an exponential integrability condition is called a branching L\'evy process, and its law is characterized by a triplet $(\sigma^2,a,\Lambda)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(\sigma^2,a,\Lambda)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred L\'evy processes conditioned to stay positive.
Fichier principal
Vignette du fichier
dermartblp.pdf (606.67 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03251327 , version 1 (26-01-2022)
hal-03251327 , version 2 (17-11-2022)

Identifiants

Citer

Bastien Mallein, Quan Shi. A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process. 2022. ⟨hal-03251327v1⟩
268 Consultations
90 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More