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Branching brownian motion with absorption

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  • Kesten, Harry

Abstract

We consider a branching diffusion {Zt}t[greater-or-equal, slanted]0 in which particles move during their life time according to a Brownian motion with drift -[mu] and variance coefficient [sigma]2, and in which each particle which enters the negative half line is instantaneously removed from the population. If particles die with probability c dt+o(dt) in [t,t+dt] and if the mean number of offspring per particle is m>1, then Zt dies out w.p.l. if [mu][greater-or-equal, slanted][mu]0[reverse not equivalent]{2[sigma]2c(m-1)}1/2. If [mu] 0} is only exp{-const.T1/3+0(logT)2}, and conditionally on {ZT>0} there are with high probability much fewer particles alive at time T than E{ZTZT0}.

Suggested Citation

  • Kesten, Harry, 1978. "Branching brownian motion with absorption," Stochastic Processes and their Applications, Elsevier, vol. 7(1), pages 9-47, March.
    • Handle: RePEc:eee:spapps:v:7:y:1978:i:1:p:9-47
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    Cited by:

    1. Huang, Chunmao & Liu, Quansheng, 2024. "Limit theorems for a branching random walk in a random or varying environment," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    2. Durrett, Rick & Mayberry, John, 2010. "Evolution in predator-prey systems," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1364-1392, July.
    3. Li, Zenghu & Zhu, Yaping, 2022. "Survival probability for super-Brownian motion with absorption," Statistics & Probability Letters, Elsevier, vol. 186(C).
    4. Liu, Jiaqi, 2021. "A Yaglom type asymptotic result for subcritical branching Brownian motion with absorption," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 245-273.
    5. Aïdékon, Elie & Jaffuel, Bruno, 2011. "Survival of branching random walks with absorption," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 1901-1937, September.

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