Abstract
We consider the model of branching random walk on an integer lattice \(\mathbb{Z}^{d}\) with periodic sources of branching. It is supposed that the regime of branching is supercritical and the Cramér condition is satisfied for a jump of the random walk. The theorem established describes the rate of front propagation for particles population over the lattice as the time increases unboundedly. The proofs are based on fundamental results related to the spatial spread of general branching random walk.
REFERENCES
M. V. Platonova and K. S. Ryadovkin, ‘‘Asymptotic behavior of the mean number of particles of branching random walk on \(\mathbf{Z}^{d}\) with periodic sources of branching,’’ Zap. Nauchn. Seminarov POMI 466, 234–256 (2017).
M. V. Platonova and K. S. Ryadovkin, ‘‘On the mean number of particles of a branching random walk on \(\mathbb{Z}^{d}\) with periodic sources of branching,’’ Dokl. Math. 97, 140–143 (2018). https://doi.org/10.1134/s1064562418020102
P. Brémaud, Markov Chains: Gibbs Fields, Monte-Carlo Simulation, and Queues, Texts in Applied Mathematics, Vol. 31 (Springer, Cham, 2020). https://doi.org/10.1007/978-3-030-45982-6
B. A. Sevast’yanov, Branching Processes (Nauka, Moscow, 1971).
J. D. Biggins, K. B. Athreya, and P. Jagers, ‘‘How fast does a general branching random walk spread?,’’ in Classical and Modern Branching Processes, Ed. by K. B. Athreya and P. Jagers (Springer, New York, 1997), pp. 19–39. https://doi.org/10.1007/978-1-4612-1862-3_2
S. A. Molchanov and E. B. Yarovaya, ‘‘Branching processes with lattice spatial dynamics and a finite set of particle generation centers,’’ Dokl. Math. 86, 638–641 (2012). https://doi.org/10.1134/s1064562412040278
E. V. Bulinskaya, ‘‘Spread of a catalytic branching random walk on a multidimensional lattice,’’ Stochastic Processes Their Appl. 128, 2325–2340 (2018). https://doi.org/10.1016/j.spa.2017.09.007
Ph. Carmona and Yu. Hu, ‘‘The spread of a catalytic branching random walk,’’ Ann. Inst. Henri Poincaré Probab. Stat. 50, 327–351 (2014). https://doi.org/10.1214/12-AIHP529
M. V. Platonova and K. S. Ryadovkin, ‘‘Branching random walks on \(\mathbf{Z}^{d}\) with periodic branching sources,’’ Theory Probab. Its Appl. 64, 229–248 (2019). https://doi.org/10.1137/S0040585X97T989465
A. A. Borovkov, Probability Theory (URSS, Moscow, 2021).
W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971).
J. D. Biggins, ‘‘The asymptotic shape of the branching random walk,’’ Adv. Appl. Probab. 10, 62–84 (1978). https://doi.org/10.1017/s0001867800029487
O. Kallenberg, Foundations of Modern Probability, Probability and Its Applications (Springer, New York, 1997). https://doi.org/10.1007/b98838
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Translated by E. Oborin
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Bulinskaya, E.V. Propagation of the Front of Random Walk with Periodic Branching Sources. Moscow Univ. Math. Bull. 79, 34–43 (2024). https://doi.org/10.3103/S0027132224700049
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DOI: https://doi.org/10.3103/S0027132224700049