Yaglom-type limit theorems for branching Brownian motion with absorption
Annales Henri Lebesgue, Volume 5 (2022), pp. 921-985.

KeywordsBranching Brownian motion, Yaglom limit theorem, continuous-state branching process

### Abstract

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.

### References

[AFGJ16] Asselah, Amine; Ferrari, Pablo A.; Groisman, Pablo; Jonckheere, Matthieu Fleming-Viot selects the minimal quasi-stationary distribution: The Galton–Watson case, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 52 (2016) no. 2, pp. 647-668 | MR | Zbl

[Ald92] Aldous, David Greedy Search on the Binary Tree with Random Edge-Weights, Comb. Probab. Comput., Volume 1 (1992) no. 4, pp. 281-293 | DOI | MR | Zbl

[BBS11] Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason Survival of near-critical branching Brownian motion, J. Stat. Phys., Volume 143 (2011) no. 5, pp. 833-854 | DOI | MR | Zbl

[BBS13] Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason The genealogy of branching Brownian motion with absorption, Ann. Probab., Volume 41 (2013) no. 2, pp. 527-618 | MR | Zbl

[BBS14] Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason Critical branching Brownian motion with absorption: survival probability, Probab. Theory Relat. Fields, Volume 160 (2014) no. 3-4, pp. 489-520 | DOI | MR | Zbl

[BBS15] Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason Critical branching Brownian motion with absorption: particle configurations, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 51 (2015) no. 4, pp. 1215-1250 | Numdam | MR | Zbl

[BDMM06] Brunet, Éric; Derrida, Bernard; Mueller, A. H.; Munier, S. Noisy traveling waves: effect of selection on genealogies, Eur. Phys. Lett., Volume 76 (2006) no. 1, pp. 1-7 | DOI | MR

[BDMM07] Brunet, Éric; Derrida, Bernard; Mueller, A. H.; Munier, S. Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization, Phys. Rev. E, Volume 76 (2007) no. 4, 041104, 20 pages | DOI | MR

[BFM08] Bertoin, Jean; Fontbona, Joaquim; Martínez, Servet On prolific individuals in a supercritical continuous-state branching process, J. Appl. Probab., Volume 45 (2008) no. 3, pp. 714-726 | DOI | MR | Zbl

[BIM20] Buraczewski, Dariusz; Iksanov, Alexander; Mallein, Bastien On the derivative martingale in a branching random walk, Ann. Probab., Volume 49 (2020) no. 3, pp. 1164-1204 | MR | Zbl

[BKMS11] Berestycki, Julien; Kyprianou, Andreas E.; Murillo-Salas, Antonio The prolific backbone for supercritical superprocesses, Stochastic Processes Appl., Volume 121 (2011) no. 6, pp. 1315-1331 | DOI | MR | Zbl

[Bra78] Bramson, Maury D. Maximal displacement of branching Brownian motion, Commun. Pure Appl. Math., Volume 31 (1978), pp. 531-581 | DOI | MR | Zbl

[Bra83] Bramson, Maury D. Convergence of solutions of the Kolmogorov equation to travelling waves, Memoirs of the American Mathematical Society, 285, American Mathematical Society, 1983 | Zbl

[CCL + 09] Cattiaux, Patrick; Collet, Pierre; Lambert, Amaury; Martínez, Servet; Méléard, Sylvie; San Martín, Jaime Quasi-stationary distributions and diffusion models in population dynamics, Ann. Probab., Volume 37 (2009) no. 5, pp. 1926-1969 | MR | Zbl

[Cha91] Chauvin, Brigitte Product martingales and stopping lines for branching Brownian motion, Ann. Probab., Volume 19 (1991) no. 3, pp. 1195-1205 | MR | Zbl

[CV16] Champagnat, Nicolas; Villemonais, Denis Exponential convergence to quasi-stationary distribution and $Q$-process, Probab. Theory Relat. Fields, Volume 164 (2016) no. 1-2, pp. 243-283 | DOI | MR | Zbl

[Dan82] Daniels, Henry E. Sequential Tests Constructed From Images, Ann. Stat., Volume 10 (1982), pp. 394-400 | MR | Zbl

[DM13] Del Moral, Pierre Mean field simulation for Monte Carlo integration, Monographs on Statistics and Applied Probability, 126, CRC Press, 2013 | DOI | Zbl

[Doo84] Doob, Joseph L. Classical Potential Theory and its Probabilistic Counterpart, Grundlehren der Mathematischen Wissenschaften, 262, Springer, 1984 | DOI | Zbl

[DS07] Derrida, Bernard; Simon, Damien The survival probability of a branching random walk in presence of an absorbing wall, Europhys. Lett., Volume 78 (2007) no. 6, 60006, 6 pages | MR | Zbl

[FM19] Foucart, Clément; Ma, Chunhua Continuous-state branching processes, extremal processes, and super-individuals, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 55 (2019) no. 2, pp. 1061-1086 | MR | Zbl

[FS04] Fleischmann, Klaus; Sturm, Anja A super-stable motion with infinite mean branching, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 40 (2004) no. 5, pp. 513-537 | DOI | Numdam | MR | Zbl

[Gar85] Gardiner, Crispin W. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer Series in Synergetics, 13, Springer, 1985 | DOI | Zbl

[GR92] Gadag, Veeresh G.; Rajarshi, Manohar B. On processes associated with a super-critical Markov branching process, Serdica, Volume 18 (1992) no. 1-4, pp. 173-178 | MR | Zbl

[Gre74] Grey, D. R. Asymptotic behavior of continuous time, continuous state-space branching processes, J. Appl. Probab., Volume 11 (1974), pp. 669-677 | DOI | Zbl

[Gre77] Grey, D. R. Almost sure convergence in Markov branching processes with infinite mean, J. Appl. Probab., Volume 14 (1977), pp. 702-716 | DOI | MR | Zbl

[Haa76] de Haan, Laurens An Abel–Tauber theorem for Laplace transforms, J. Lond. Math. Soc., Volume 13 (1976), pp. 537-542 | DOI | MR | Zbl

[HH07] Harris, John W.; Harris, Simon C. Survival probabilities for branching Brownian motion with absorption, Electron. Commun. Probab., Volume 12 (2007), pp. 81-92 | MR | Zbl

[HHK06] Harris, John W.; Harris, Simon C.; Kyprianou, Andreas E. Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: one-sided travelling waves, Ann. Inst. H. Poincaré Probab. Stat., Volume 42 (2006) no. 1, pp. 125-145 | DOI | Numdam | MR | Zbl

[IM74] Itô, Kiyosi; McKean, Henry P. Jr. Diffusion Processes and Their Sample Paths, Grundlehren der Mathematischen Wissenschaften, 125, Springer, 1974 | Zbl

[INW69] Ikeda, Nobuyuki; Nagasawa, Masao; Watanabe, Shinzo Branching Markov Processes. III, J. Math. Kyoto Univ., Volume 9 (1969), pp. 95-160 | MR | Zbl

[Kes78] Kesten, Harry Branching Brownian motion with absorption, Stochastic Processes Appl., Volume 7 (1978), pp. 9-47 | DOI | MR | Zbl

[KPP37] Kolmogorov, Andreĭ; Petrovskiĭ, Ivan; Piscounov, Nikolaĭ Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, Sér. Int., Sect. A: Math. et Mécan, Volume 1 (1937) no. 6, pp. 1-25 | Zbl

[KS91] Karatzas, Ioannis; Shreve, Steven E. Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113, Springer, 1991 | Zbl

[Kyp04] Kyprianou, Andreas E. Travelling wave solutions to the K-P-P equation: Alternatives to Simon Harris’ probabilistic analysis, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 40 (2004) no. 1, pp. 53-72 | DOI | Numdam | MR | Zbl

[Law06] Lawler, Gregory F. Introduction to Stochastic Processes, Chapman & Hall/CRC, 2006 | Zbl

[Ler86] Lerche, Hans R. Boundary crossing of Brownian motion, Lecture Notes in Statistics, 40, Springer, 1986 | DOI | MR | Zbl

[LPP95] Lyons, Russell; Pemantle, Robin; Peres, Yuval Conceptual proofs of $L$ Log $L$ criteria for mean behavior of branching processes, Ann. Probab., Volume 23 (1995) no. 3, pp. 1125-1138 | Zbl

[Mai12] Maillard, Pascal Branching Brownian motion with selection, Ph. D. Thesis, Université Pierre et Marie Curie, Paris, France (2012) (https://arxiv.org/abs/1210.3500v1)

[Mai16] Maillard, Pascal Speed and fluctuations of $N$-particle branching Brownian motion with spatial selection, Probab. Theory Relat. Fields, Volume 166 (2016) no. 3-4, pp. 1061-1173 | DOI | MR | Zbl

[McK75] McKean, Henry P. Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov, Commun. Pure Appl. Math., Volume 28 (1975), pp. 323-331 | DOI | MR | Zbl

[MR21] Mallein, Bastien; Ramassamy, Sanjay Barak–Erdős graphs and the infinite-bin model, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 57 (2021) no. 4, pp. 1940-1967 | Zbl

[MV12] Méléard, Sylvie; Villemonais, Denis Quasi-stationary distributions and population processes, Probab. Surv., Volume 9 (2012), pp. 340-410 | MR | Zbl

[Nev] Neveu, Jacques A continuous-state branching process in relation with the GREM model of spin glass theory (Rapport interne 267, École polytechnique)

[Nev88] Neveu, Jacques Multiplicative martingales for spatial branching processes, Seminar on Stochastic Processes, 1987 (Progress in Probability and Statistics), Volume 15, Birkhäuser, 1988, pp. 223-241 | DOI | MR | Zbl

[Nov81] Novikov, Aleksandr A. On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary, Math. USSR, Sb., Volume 38 (1981), pp. 495-505 | DOI | Zbl

[Rob15] Roberts, Matthew I. Fine asymptotics for the consistent maximal displacement of branching Brownian motion, Electron. J. Probab., Volume 20 (2015), 28 | MR | Zbl

[Yag47] Yaglom, Akiva M. Certain limit theorems of the theory of branching random processes, Dokl. Akad. Nauk SSSR, n. Ser., Volume 56 (1947), pp. 795-798 | MR | Zbl