Abstract
We consider a symmetric branching random walk in a multi-dimensional lattice with continuous time and Markov branching process at each lattice point. It is assumed that initially at each lattice point there is one particle and in the process of branching any particle can produce an arbitrary number of descendants. For a critical process, under the assumption that the walk is transient, we prove the convergence of the distribution of the particle field to the limit stationary distribution. We show the absence of intermittency in the zone \(|x-y| = O(\sqrt{t})\), where \(x\) and \(y\) are spatial coordinates and \(t\) is the time, under the assumption of superexponentially light tails of a random walk and a supercriticality of the branching process at the points of the lattice.
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D. Balashova, S. Molchanov, and E. Yarovaya, “Structure of the particle population for a branching random walk with a critical reproduction law,” Methodol. Comput. Appl. Probab. 23 (1), 85–102 (2021).
A. A. Borovkov, Asymptotic Analysis of Random Walks: Light-Tailed Distributions (Cambridge Univ. Press, Cambridge, 2020), Encycl. Math. Appl. 176 [transl. from the Russian (Fizmatlit, Moscow, 2013)].
E. Chernousova, Y. Feng, O. Hryniv, S. Molchanov, and J. Whitmeyer, “Steady states of lattice population models with immigration,” Math. Popul. Stud. 28 (2), 63–80 (2021).
E. Chernousova, O. Hryniv, and S. Molchanov, “Population model with immigration in continuous space,” Math. Popul. Stud. 27 (4), 199–215 (2020).
E. Chernousova and S. Molchanov, “Steady state and intermittency in the critical branching random walk with arbitrary total number of offspring,” Math. Popul. Stud. 26 (1), 47–63 (2019).
E. Ermakova, P. Makhmutova, and E. Yarovaya, “Branching random walks and their applications for epidemic modeling,” Stoch. Models 35 (3), 300–317 (2019).
W. Feller, An Introduction to Probability Theory and Its Applications (J. Wiley & Sons, New York, 2008), Vol. 2.
A. Getan, S. Molchanov, and B. Vainberg, “Intermittency for branching walks with heavy tails,” Stoch. Dyn. 17 (6), 1750044 (2017).
I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes (Nauka, Moscow, 1977) [in Russian].
S. Molchanov and J. Whitmeyer, “Stationary distributions in Kolmogorov–Petrovski–Piskunov-type models with an infinite number of particles,” Math. Popul. Stud. 24 (3), 147–160 (2017).
S. A. Molchanov and E. B. Yarovaya, “Limit theorems for the Green function of the lattice Laplacian under large deviations of the random walk,” Izv. Math. 76 (6), 1190–1217 (2012) [transl. from Izv. Ross. Akad. Nauk, Ser. Mat. 76 (6), 123–152 (2012)].
S. A. Molchanov and E. B. Yarovaya, “Large deviations for a symmetric branching random walk on a multidimensional lattice,” Proc. Steklov Inst. Math. 282, 186–201 (2013) [transl. from Tr. Mat. Inst. Steklova 282, 195–211 (2013)].
J. A. Shohat and J. D. Tamarkin, The Problem of Moments (Am. Math. Soc., New York, 1943), Math. Surv. 1.
V. A. Vatutin and A. M. Zubkov, “Branching processes. I,” J. Sov. Math. 39 (1), 2431–2475 (1987) [transl. from Itogi Nauki Tekh., Ser.: Teor. Veroyatn., Mat. Stat., Teor. Kibern. 23, 3–67 (1985)].
V. A. Vatutin and A. M. Zubkov, “Branching processes. II,” J. Sov. Math. 67 (6), 3407–3485 (1993).
E. B. Yarovaya, Branching Random Walks in an Inhomogeneous Medium (Tsentr Prikl. Issled. Mekh.-Mat. Fak. Mosk. Gos. Univ., Moscow, 2007) [in Russian].
E. B. Yarovaya, “The monotonicity of the probability of return into the source in models of branching random walks,” Moscow Univ. Math. Bull. 65 (2), 78–80 (2010) [transl. from Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No. 2, 44–47 (2010)].
E. B. Yarovaya, “Models of branching walks and their use in the reliability theory,” Autom. Remote Control 71 (7), 1308–1324 (2010) [transl. from Avtom. Telemekh., No. 7, 29–46 (2010)].
E. B. Yarovaya, J. M. Stoyanov, and K. K. Kostyashin, “On conditions for a probability distribution to be uniquely determined by its moments,” Theory Probab. Appl. 64 (4), 579–594 (2020) [transl. from Teor. Veroyatn. Primen. 64 (4), 725–745 (2019)].
Acknowledgments
The authors are grateful to Prof. S. A. Molchanov for useful discussions and to the referee for comments that helped to improve the presentation.
Funding
This work is supported by the Russian Foundation for Basic Research, project no. 20-01-00487.
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 316, pp. 64–78 https://doi.org/10.4213/tm4248.
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Balashova, D.M., Yarovaya, E.B. Structure of the Population of Particles for a Branching Random Walk in a Homogeneous Environment. Proc. Steklov Inst. Math. 316, 57–71 (2022). https://doi.org/10.1134/S0081543822010060
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DOI: https://doi.org/10.1134/S0081543822010060