Consider a branching random walk with a mechanism of elimination. We assume that the underlying Galton-Watson process is supercritical, thus the branching random walk has a positive survival probability. A mechanism of elimination, which is called a barrier, is introduced to erase the particles who lie above $ ri+\varepsilon i^{\alpha} $ and all their descendants, where $ i $ presents the generation of the particles, $ \alpha > 1/3, \varepsilon\in{{\mathbb{R}}} $ and $ r $ is the asymptotic speed of the left-most position of the branching random walk. First we show that the particle system still has a positive survival probability after we introduce the barrier with $ \varepsilon > 0. $ Moreover, we show that the decay of the probability is faster than $ e^{-\beta'\varepsilon^{\beta}} $ as $ \varepsilon\downarrow 0 $, where $ \beta', \beta $ are two positive constants depending on the branching random walk and $ \alpha $. The result in the present paper extends a conclusion in Gantert et al. (2011) in some extent. Our proof also works for some time-inhomogeneous cases.
Citation: You Lv. Asymptotic behavior of survival probability for a branching random walk with a barrier[J]. AIMS Mathematics, 2023, 8(2): 5049-5059. doi: 10.3934/math.2023253
Consider a branching random walk with a mechanism of elimination. We assume that the underlying Galton-Watson process is supercritical, thus the branching random walk has a positive survival probability. A mechanism of elimination, which is called a barrier, is introduced to erase the particles who lie above $ ri+\varepsilon i^{\alpha} $ and all their descendants, where $ i $ presents the generation of the particles, $ \alpha > 1/3, \varepsilon\in{{\mathbb{R}}} $ and $ r $ is the asymptotic speed of the left-most position of the branching random walk. First we show that the particle system still has a positive survival probability after we introduce the barrier with $ \varepsilon > 0. $ Moreover, we show that the decay of the probability is faster than $ e^{-\beta'\varepsilon^{\beta}} $ as $ \varepsilon\downarrow 0 $, where $ \beta', \beta $ are two positive constants depending on the branching random walk and $ \alpha $. The result in the present paper extends a conclusion in Gantert et al. (2011) in some extent. Our proof also works for some time-inhomogeneous cases.
| [1] |
J. M. Hammersley, Postulates for subadditive processes, Ann. Probab., 2 (1974), 652–680. http://doi.org/10.1214/aop/1176996611 doi: 10.1214/aop/1176996611
|
| [2] |
J. F. C. Kingman, The first birth problem for an age dependent branching process, Ann. Probab., 3 (1975), 790–801. http://doi.org/10.1214/aop/1176996266 doi: 10.1214/aop/1176996266
|
| [3] |
J. D. Biggins, The first and last birth problems for a multitype age-dependent branching process, Adv. Appl. Probab., 8 (1976), 446–459. http://doi.org/10.1017/S0001867800042348 doi: 10.1017/S0001867800042348
|
| [4] | B. Lubachevsky, A. Shwartz, A. Weiss, The stability of branching random walks with a barrier, Israel: EE PUB 748, Technion, 1990. |
| [5] |
B. Lubachevsky, A. Shwartz, A. Weiss, An analysis of rollback-based simulation, ACM Trans. Model. Comput. Simulat., 1 (1991), 154–193. http://doi.org/10.1145/116890.116912 doi: 10.1145/116890.116912
|
| [6] |
J. D. Biggins, B. D. Lubachevsky, A. Shwartz, A. Weiss, A branching random walk with a barrier, Ann. Appl. Probab., 1 (1991), 573–581. http://doi.org/10.1214/aoap/1177005839 doi: 10.1214/aoap/1177005839
|
| [7] |
B. Jaffuel, The critical barrier for the survival of branching random walk with absorption, Ann. Inst. H. Poincaré Probab. Statist., 48 (2012), 989–1009. http://doi.org/10.1214/11-AIHP453 doi: 10.1214/11-AIHP453
|
| [8] |
N. Gantert, Y. Hu, Z. Shi, Asymptotics for the survival probability in a killed branching random walk, Ann. Inst. H. Poincaré Probab. Statist., 47 (2011), 111–129. http://doi.org/10.1214/10-AIHP362 doi: 10.1214/10-AIHP362
|
| [9] |
M. Fang, O. Zeitouni, Branching random walks in time inhomogeneous environments, Electron. J. Probab., 17 (2012), 1–18. http://doi.org/10.1214/EJP.v17-2253 doi: 10.1214/EJP.v17-2253
|
| [10] |
B. Mallein, Maximal displacement in a branching random walk through interfaces, Electron. J. Probab., 20 (2015), 1–40. http://doi.org/10.1214/EJP.v20-2828 doi: 10.1214/EJP.v20-2828
|
| [11] |
B. Mallein, Maximal displacement of a branching random walk in time-inhomogeneous environment, Stoch. Proc. Appl., 125 (2015), 3958–4019. http://doi.org/10.1016/j.spa.2015.05.011 doi: 10.1016/j.spa.2015.05.011
|
| [12] | J. Peyriére, Turbulence et dimension de Hausdorff, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 567–569. |
| [13] |
J. P. Kahane, J. Peyriére, Sur certaines martingales de Benoit Mandelbrot, Adv. Math., 22 (1976), 131–145. http://doi.org/10.1016/0001-8708(76)90151-1 doi: 10.1016/0001-8708(76)90151-1
|
| [14] |
J. D. Biggins, A. E. Kyprianou, Measure change in multitype branching, Adv. Appl. Probab., 36 (2004), 544–581. http://doi.org/10.1239/aap/1086957585 doi: 10.1239/aap/1086957585
|
| [15] |
A. A. Mogul'skiĭ, Small deviations in the space of trajectories, Theor. Probab. Appl., 19 (1974), 726–736. http://doi.org/10.1137/1119081 doi: 10.1137/1119081
|
| [16] | A. A. Borovkov, A. A. Mogul'skiǐ, On probabilities of small deviations for stochastic processes, Sib. Adv. Math., 1 (1991), 39–63. |
| [17] |
Q. M. Shao, A small deviation theorem for independent random variables, Theor. Probab. Appl., 40 (1995), 225–235. http://doi.org/10.1137/1140021 doi: 10.1137/1140021
|
| [18] | Y. Lv, W. Hong, Quenched small deviation for the trajectory of a random walk with time-inhomogeneous random environment, Theor. Probab. Appl., in press. |
You Lv. Asymptotic behavior of survival probability for a branching random walk with a barrier[J]. AIMS Mathematics, 2023, 8(2): 5049-5059. doi: 10.3934/math.2023253
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