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.
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References
Gärtner J, Molchanov SA (1990) Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm Math Phys 132 (3):613–655. http://projecteuclid.org/getRecord?id=euclid.cmp/1104201232
Gikhman II, Skorokhod AV (1969) Introduction to the theory of random processes. Translated from the Russian by Scripta Technica, Inc. W. B. Saunders Co., Philadelphia
Kondratiev Y, Kutoviy O, Minlos R, Pirogov S (2013) On spatial mutation-selection models. J Math Phys 54(11):113,504, 13. https://doi.org/10.1063/1.4828856
Molchanov S, Vainberg B (2018) Population dynamics with moderate tails of the underlying random walk. ArXiv:1805.03133
Molchanov S, Whitmeyer J (2017) Stationary distributions in Kolmogorov-Petrovski-Piskunov-type models with an infinite number of particles. Math Popul Stud 24(3):147–160. https://doi.org/10.1080/08898480.2017.1330010
Molchanov SA, Yarovaya EB (2013) Large deviations for a symmetric branching random walk on a multidimensional lattice. Tr Mat Inst Steklova 282(Vetvyashchiesya Protsessy, Sluchaı̆nye Bluzhdaniya, i Smezhnye Voprosy):195–211. https://doi.org/10.1134/s0081543813060163
Zeldovich YB, Molchanov SA, Ruzmaikin AA, Sokoloff DD (2014) Intermittency, diffusion and generation in a nonstationary random medium. Reviews in mathematics and mathematical physics, vol 15/1. Cambridge Scientific Publishers, Cambridge. second edition of [ MR1128327]
Acknowledgements
D. Balashova and E. Yarovaya were supported by the Russian Foundation for Basic Research (RFBR), project No. 17-01-00468. S. Molchanov was supported by the Russian Science Foundation (RSF), project No. 17-11-01098.
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Balashova, D., Molchanov, S. & Yarovaya, E. Structure of the Particle Population for a Branching Random Walk with a Critical Reproduction Law. Methodol Comput Appl Probab 23, 85–102 (2021). https://doi.org/10.1007/s11009-020-09773-2
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DOI: https://doi.org/10.1007/s11009-020-09773-2