Critical branching brownian motion with absorption: Particle configurations
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, p. 1215-1250
Nous considérons un mouvement brownien branchant avec absorption critique, issu d’une particule en x>0, dans lequel les particules se déplacent selon des mouvement browniens réels indépendants avec une dérive critique de -2, et sont absorbées en zero. Nous obtenons des résultats asymptotiques sur le comportement de ce processus avant son extinction, quand la position x de la particule initiale tend vers l’infini. En particulier nous obtenons des éstimées sur le nombre de particules dans le système, la position de la particule la plus à droite, et la configuration des particules à un instant typique.
We consider critical branching Brownian motion with absorption, in which there is initially a single particle at x>0, particles move according to independent one-dimensional Brownian motions with the critical drift of -2, and particles are absorbed when they reach zero. Here we obtain asymptotic results concerning the behavior of the process before the extinction time, as the position x of the initial particle tends to infinity. We estimate the number of particles in the system at a given time and the position of the right-most particle. We also obtain asymptotic results for the configuration of particles at a typical time.
@article{AIHPB_2015__51_4_1215_0,
     author = {Berestycki, Julien and Berestycki, Nathana\"el and Schweinsberg, Jason},
     title = {Critical branching brownian motion with absorption: Particle configurations},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {4},
     year = {2015},
     pages = {1215-1250},
     doi = {10.1214/14-AIHP613},
     zbl = {1329.60300},
     mrnumber = {3414446},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_4_1215_0}
}
Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason. Critical branching brownian motion with absorption: Particle configurations. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1215-1250. doi : 10.1214/14-AIHP613. https://www.numdam.org/item/AIHPB_2015__51_4_1215_0/

[1] E. Aïdékon, J. Berestycki, E. Brunet and Z. Shi. Branching Brownian motion seen from its tip. Probab. Theory Related Fields 157 (2013) 405–451. | MR 3101852 | Zbl 1284.60154

[2] L.-P. Arguin, A. Bovier and N. Kistler. The extremal process of branching Brownian motion. Probab. Theory Related Fields 157 (2013) 535–574. | MR 3129797 | Zbl 1286.60045

[3] A. Asselah, P. Ferrari and P. Groisman. Quasi-stationary distributions and Fleming–Viot processes in finite spaces. J. Appl. Probab. 48 (2011) 322–332. | MR 2840302 | Zbl 1219.60081

[4] A. Asselah, P. Ferrari, P. Groisman and M. Jonckheere. Fleming–Viot selects the minimal quasi-stationary distribution: The Galton–Watson case. Ann. Inst. Henri Poincaré. To appear, 2015. Available at arXiv:1206.6114.

[5] J. Berestycki, N. Berestycki and J. Schweinsberg. The genealogy of branching Brownian motion with absorption. Ann. Probab. 41 (2013) 527–618. | MR 3077519 | Zbl 1304.60088

[6] J. Berestycki, N. Berestycki and J. Schweinsberg. Critical branching Brownian motion with absorption: Survival probability. Probab. Theory Related Fields 160 (2014) 489–520. | MR 3278914 | Zbl 06380858

[7] M. Bramson. Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44 (285) (1983) iv+190. | MR 705746 | Zbl 0517.60083

[8] E. Brunet, B. Derrida, A. H. Mueller and S. Munier. Noisy traveling waves: Effect of selection on genealogies. Europhys. Lett. 76 (2006) 1–7. | MR 2299937

[9] E. Brunet, B. Derrida, A. H. Mueller and S. Munier. Effect of selection on ancestry: An exactly soluble case and its phenomenological generalization. Phys. Rev. E (3) 76 (2007) 041104. | MR 2365627

[10] K. Burdzy, R. Holyst, D. Ingerman and P. March. Configurational transition in a Fleming–Viot-type model and probabilistic interpretation of Laplacian eigenfunctions. J. Phys. A: Math. Gen. 29 (1996) 2633–2642. | Zbl 0901.60054

[11] K. Burdzy, R. Holyst and P. March. A Fleming–Viot particle representation of the Dirichlet Laplacian. Comm. Math. Phys. 214 (2000) 679–703. | MR 1800866 | Zbl 0982.60078

[12] I. Grigorescu and M. Kang. Hydrodynamic limit for a Fleming–Viot type system. Stochastic Process. Appl. 110 (2004) 111–143. | MR 2052139 | Zbl 1075.60124

[13] I. Grigorescu and M. Kang. Immortal particle for a catalytic branching process. Probab. Theory Related Fields 153 (2011) 333–361. | MR 2925577 | Zbl 1251.60064

[14] F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik. A short proof of the logarithmic Bramson correction in Fisher–KPP equations. Netw. Heterog. Media 8 (2013) 275–289. | MR 3043938 | Zbl 1275.35067

[15] F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik. The logarithmic delay of KPP fronts in a periodic medium. Preprint, arXiv:1211.6173.

[16] J. W. Harris and S. C. Harris. Survival probabilities for branching Brownian motion with absorption. Electron. Commun. Probab. 12 (2007) 81–92. | MR 2300218 | Zbl 1132.60059

[17] J. W. Harris, S. C. Harris and A. E. Kyprianou. Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: One-sided traveling waves. Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 125–145. | Numdam | MR 2196975 | Zbl 1093.60059

[18] S. C. Harris and M. I. Roberts. The unscaled paths of branching Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 579–608. | Numdam | MR 2954267 | Zbl 1259.60102

[19] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Springer, New York, 2002. | MR 1876169 | Zbl 0892.60001

[20] H. Kesten. Branching Brownian motion with absorption. Stochastic Process. Appl. 7 (1978) 9–47. | MR 494543 | Zbl 0383.60077

[21] S. P. Lalley and Y. Shao. On the maximal displacement of a critical branching random walk. Probab. Theory Related Fields 162 (2015) 71–96. | MR 3350041

[22] P. Maillard. Speed and fluctuations of N-particle branching Brownian motion with spatial selection. Preprint, arXiv:1304.0562.

[23] J. Neveu. Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes, 1987 223–241. E. Çinlar, K. L. Chung and R. K. Getoor (Eds). Prog. Probab. Statist. 15. Birkhäuser, Boston, 1988. | MR 1046418 | Zbl 0652.60089

[24] S. Sawyer. Branching diffusion processes in population genetics. Adv. in Appl. Probab. 8 (1976) 659–689. | MR 432250 | Zbl 0365.60081

[25] S. Martinez and J. San Martin. Quasi-stationary distributions for a Brownian motion with drift and associated limit laws. J. Appl. Probab. 31 (4) (1994) 911–920. | MR 1303922 | Zbl 0818.60071

[26] A. M. Yaglom. Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.) 56 (1947) 795–798. | MR 22045 | Zbl 0041.45602