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The quasi-stationary distribution for small random perturbations of certain one-dimensional maps

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  • Ramanan, Kavita
  • Zeitouni, Ofer

Abstract

We analyse the quasi-stationary distributions of the family of Markov chains {Xn[var epsilon]},[var epsilon]>0, obtained from small non-local random perturbations of iterates of a map f : I-->I on a compact interval. The class of maps considered is slightly more general than the class of one-dimensional Axiom A maps. Under certain conditions on the dynamics, we show that as [var epsilon]-->0 the limit quasi-stationary distribution of the family of Markov chains is supported on the union of the periodic attractors of the map f. Moreover, we show that these conditions are satisfied by Markov chains obtained as perturbations of the logistic map f(x)=[mu]x(1-x) by additive Gaussian noise and also by Markov chains that model density-dependent branching processes.

Suggested Citation

  • Ramanan, Kavita & Zeitouni, Ofer, 1999. "The quasi-stationary distribution for small random perturbations of certain one-dimensional maps," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 25-51, November.
    • Handle: RePEc:eee:spapps:v:84:y:1999:i:1:p:25-51
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    References listed on IDEAS

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    1. Klebaner, F. C. & Nerman, O., 1994. "Autoregressive approximation in branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 51(1), pages 1-7, June.
    2. Klebaner, Fima C., 1993. "Population-dependent branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 46(1), pages 115-127, May.
    3. Högnäs, Göran, 1997. "On the quasi-stationary distribution of a stochastic Ricker model," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 243-263, October.
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    Cited by:

    1. Mao, Xuerong & Marion, Glenn & Renshaw, Eric, 2002. "Environmental Brownian noise suppresses explosions in population dynamics," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 95-110, January.
    2. Tian, Rongrong & Wei, Jinlong & Wu, Jiang-Lun, 2021. "On a generalized population dynamics equation with environmental noise," Statistics & Probability Letters, Elsevier, vol. 168(C).

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