Abstract
Branching random walks play a key role in modeling the evolutionary processes with birth and death of particles depending on the structure of a medium. The branching random walk on a multidimensional lattice with a finite number of branching sources of three types is investigated. It is assumed that the intensities of branching in the sources can be arbitrary. The principal attention is paid to the analysis of spectral characteristics of the operator describing evolution of the mean numbers of particles both at an arbitrary point and on the entire lattice. The obtained results provide an explicit conditions for the exponential growth of the numbers of particles without any assumptions on jumps variance of the underlying random walk.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Albeverio, S., Bogachev, L.V., Yarovaya, E.B.: Asymptotics of branching symmetric random walk on the lattice with a single source. C. R. Acad. Sci. Paris Sér. I Math. 326(8), 975–980 (1998). https://doi.org/10.1016/S0764-4442(98)80125-0
Bogachev, L.V., Yarovaya, E.B.: A limit theorem for a supercritical branching random walk on \(\mathbf{Z}^{d}\) with a single source. Russian Math. Surv. 53(5), 1086–1088 (1998). https://doi.org/10.1070/rm1998v053n05ABEH000077
Gikhman, I.I., Skorokhod, A.V.: The Theory of Stochastic Processes. II. Classics in Mathematics. Springer, Berlin (2004).Translated from the Russian by S. Kotz, Reprint of the 1975 edition
Halmos, P.R.: A Hilbert Space Problem Book. Graduate Texts in Mathematics, vol. 19, 2nd edn. Springer, New York (1982). https://doi.org/10.1007/978-1-4684-9330-6Encyclopedia of Mathematics and its Applications, 17
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985). https://doi.org/10.1017/CBO9780511810817
Kato, T.: Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer, New York (1966). https://doi.org/10.1007/978-3-662-12678-3
Khristolyubov, I.I., Yarovaya, E.B.: A limit theorem for a supercritical branching walk with sources of varying intensity. Teor. Veroyatn. Primen. 64(3), 456–480 (2019). https://doi.org/10.1137/S0040585X97T989556
Shohat, J.A., Tamarkin, J.D.: The Problem of Moments. American Mathematical Society Mathematical Surveys, vol. I. American Mathematical Society, New York (1943)
Vatutin, V.A., Topchiĭ, V.A., Yarovaya, E.B.: Catalytic branching random walks and queueing systems with a random number of independent servers. Teor. Ĭmovīr. Mat. Stat. 69, 1–15 (2003)
Yarovaya, E.: Spectral asymptotics of a supercritical branching random walk. Theory Probab. Appl. 62(3), 413–431 (2018). https://doi.org/10.1137/S0040585X97T98871X
Yarovaya, E.B.: Branching random walks in a heterogeneous environment. Center of Applied Investigations of the Faculty of Mechanics and Mathematics of the Moscow State University, Moscow (2007). (in Russian)
Yarovaya, E.B.: Spectral properties of evolutionary operators in branching random walk models. Math. Notes 92(1–2), 115–131 (2012). Translation of Mat. Zametki 92 (2012), no. 1, 123–140. https://doi.org/10.1134/S0001434612070139
Yarovaya, E.B.: Branching random walks with several sources. Math. Popul. Stud. 20(1), 14–26 (2013). https://doi.org/10.1080/08898480.2013.748571
Yarovaya, E.: Positive discrete spectrum of the evolutionary operator of supercritical branching walks with heavy tails. Methodol. Comput. Appl. Probab. 19(4), 1151–1167 (2016). https://doi.org/10.1007/s11009-016-9492-9
Acknowledgment
The research was supported by the Russian Foundation for the Basic Research (RFBR), project No. 20-01-00487.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Yarovaya, E., Balashova, D., Khristolyubov, I. (2021). Branching Walks with a Finite Set of Branching Sources and Pseudo-sources. In: Shiryaev, A.N., Samouylov, K.E., Kozyrev, D.V. (eds) Recent Developments in Stochastic Methods and Applications. ICSM-5 2020. Springer Proceedings in Mathematics & Statistics, vol 371. Springer, Cham. https://doi.org/10.1007/978-3-030-83266-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-83266-7_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-83265-0
Online ISBN: 978-3-030-83266-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)