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Structure of the Particle Population for a Branching Random Walk with a Critical Reproduction Law

Author

Listed:
  • Daria Balashova

    (Lomonosov Moscow State University)

  • Stanislav Molchanov

    (University of North Carolina at Charlotte
    National Research University Higher School of Economics)

  • Elena Yarovaya

    (Lomonosov Moscow State University)

Abstract

We consider a continuous-time symmetric branching random walk on the d-dimensional lattice, d ≥ 1, and assume that at the initial moment there is one particle at every lattice point. Moreover, we assume that the underlying random walk has a finite variance of jumps and the reproduction law is described by a continuous-time Markov branching process (a continuous-time analog of a Bienamye-Galton-Watson process) at every lattice point. We study the structure of the particle subpopulation generated by the initial particle situated at a lattice point x. We replay why vanishing of the majority of subpopulations does not affect the convergence to the steady state and leads to clusterization for lattice dimensions d = 1 and d = 2.

Suggested Citation

  • Daria Balashova & Stanislav Molchanov & Elena Yarovaya, 2021. "Structure of the Particle Population for a Branching Random Walk with a Critical Reproduction Law," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 85-102, March.
  • Handle: RePEc:spr:metcap:v:23:y:2021:i:1:d:10.1007_s11009-020-09773-2
    DOI: 10.1007/s11009-020-09773-2
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    References listed on IDEAS

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    1. Stanislav Molchanov & Joseph Whitmeyer, 2017. "Stationary distributions in Kolmogorov-Petrovski- Piskunov-type models with an infinite number of particles," Mathematical Population Studies, Taylor & Francis Journals, vol. 24(3), pages 147-160, July.
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    Cited by:

    1. Iuliia Makarova & Daria Balashova & Stanislav Molchanov & Elena Yarovaya, 2022. "Branching Random Walks with Two Types of Particles on Multidimensional Lattices," Mathematics, MDPI, vol. 10(6), pages 1-45, March.

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    1. Yaqin Feng & Stanislav Molchanov & Elena Yarovaya, 2021. "Stability and Instability of Steady States for a Branching Random Walk," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 207-218, March.

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