Fleming-Viot selects the minimal quasi-stationary distribution: the Galton-Watson case. (English. French summary) Zbl 1342.60145

Summary: Consider \(N\) particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits \(0\), at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each \(N\) there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as \(N\) goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.


60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
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[1] A. Asselah, P. A. Ferrari and P. Groisman. Quasi-stationary distributions and Fleming-Viot processes in finite spaces. J. Appl. Probab. 48 (2) (2011) 322-332. · Zbl 1219.60081
[2] J. Bérard and J. B. Gouéré. Brunet-Derrida behavior of branching-selection particles systems on the line. Comm. Math. Phys. 298 (2) (2010) 323-342. · Zbl 1247.60124
[3] J. Berestycki, N. Berestycki and J. Schweinsberg. The genealogy of branching Brownian motion with absorption. Ann. Probab. 41 (2) (2013) 527-618. · Zbl 1304.60088
[4] M. Bieniek, K. Burdzy and S. Finch. Non-extinction of a Fleming-Viot particle model. Probab. Theory Related Fields 153 (1-2) 293-332. · Zbl 1253.60089
[5] E. Brunet and B. Derrida. Effect of microscopic noise on front propagation. J. Stat. Phys. 103 (1-2) (2001) 269-282. · Zbl 1018.82020
[6] E. Brunet and B. Derrida. Shift in the velocity of a front due to a cutoff. Phys. Rev. E (3) 56 (3) (1997), part A, 2597-2604.
[7] E. Brunet, B. Derrida, A. H. Mueller and S. Munier. Noisy traveling waves: Effect of selection on genealogies. Europhys. Lett. 76 (1) (2006) 1-7.
[8] 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 (4) (2007) 1-20.
[9] 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
[10] J. A. Cavender. Quasi-stationary distributions of birth-and-death processes. Adv. in Appl. Probab. 10 (3) (1978) 570-586. · Zbl 0381.60068
[11] R. Durrett and D. Remenik. Brunet-Derrida particles systems, free boundary problems and Wiener-Hopf equations. Ann. Probab. 39 (6) (2011) 2043-2078. · Zbl 1243.60066
[12] S. N. Ethier and T. G. Kurtz. Markov Processes. Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics . Wiley, New York, 1986. · Zbl 0592.60049
[13] P. A. Ferrari, H. Kesten, S. Martinez and P. Picco. Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Probab. 23 (2) (1995) 501-521. · Zbl 0827.60061
[14] P. A. Ferrari and N. Maric. Quasi-stationary distributions and Fleming-Viot processes in countable spaces. Electron. J. Probab. 12 (24) (2007) 684-702. · Zbl 1127.60088
[15] W. H. Fleming and M. Viot. Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28 (5) (1979) 817-843. · Zbl 0444.60064
[16] I. Grigorescu and M. Kang. Hydrodynamic limit for a Fleming-Viot type system. Stochastic Process. Appl. 110 (1) (2004) 111-143. · Zbl 1075.60124
[17] I. Grigorescu and M. Kang. Immortal particle for a catalytic branching process. Probab. Theory Related Fields 153 (1-2) (2011) 333-361. · Zbl 1251.60064
[18] S. C. Harris and M. I. Roberts. The many-to-few lemma and multiple spines. Available at . arXiv:1106.4761
[19] P. Maillard. Branching Brownian motion with selection of the \(N\) right-most particles: An approximate model. Available at . arXiv:1112.0266v2 · Zbl 0273.22014
[20] M. K. Nakayama, P. Shahabuddin and K. Sigman. On finite exponential moments for branching processes and busy periods for queues. J. Appl. Probab. 41 (2004). · Zbl 1056.60085
[21] P. Robert. Stochastic Networks and Queues. Stochastic Modelling and Applied Probability. Applications of Mathematics 52 . Springer, New York, 2003.
[22] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales, 1: Foundations, 2nd edition. Wiley, Chichester, 1994. · Zbl 0826.60002
[23] E. Seneta and D. Vere-Jones. On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab. 3 (1966) 403-434. · Zbl 0147.36603
[24] D. Villemonais. Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift. Electron. J. Probab. 16 (2011) 1663-1692. · Zbl 1244.82052
[25] A. M. Yaglom. Certain limit theorems of the theory of branching random processes. Dokl. Akad. Nauk SSSR (N.S.) 56 (1947) 795-798.
[26] V. M. Zolotarev. More exact statements of several theorems in the theory of branching processes. Theory Probab. Appl. 2 (3) (1957) 245-253. · Zbl 0089.34202
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