Yaglom-type limit theorems for branching Brownian motion with absorption

Maillard, Pascal; Schweinsberg, Jason

Annales Henri Lebesgue, Tome 5 (2022), pp. 921-985.

Nous considérons un mouvement brownien branchant unidimensionnel dans lequel les particules sont absorbées à l’origine. Nous supposons que le branchement est surcritique, mais les particules reçoivent une dérive critique vers l’origine de sorte que le processus finit par s’éteindre presque sûrement. Nous établissons des asymptotiques précises pour la probabilité que le processus survive pendant un grand temps $t$, en nous appuyant sur les résultats précédents de Kesten (1978) et Berestycki, Berestycki et Schweinsberg (2014). Nous prouvons également un théorème limite de type Yaglom pour le comportement du processus conditionné pour survivre pendant un temps inhabituellement long, apportant ainsi une réponse essentiellement complète à une question soulevée par Kesten (1978). Un outil important dans les preuves de ces résultats est la convergence d’une certaine observable vers un processus de branchement à état continu. Nos preuves incorporent de nouvelles idées qui pourraient être utiles dans d’autres modèles de branchement.

We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time $t$, building on previous results by Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, providing an essentially complete answer to a question first addressed by Kesten (1978). An important tool in the proofs of these results is the convergence of a certain observable to a continuous state branching process. Our proofs incorporate new ideas which might be of use in other branching models.

Classification : 60J80, 60J65, 60J25
Mots clés : Branching Brownian motion, Yaglom limit theorem, continuous-state branching process

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