Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T14:25:31.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Heavy-tailed branching random walks on multidimensional lattices. A moment approach

Part of: Markov processes

Published online by Cambridge University Press:  22 July 2020

Anastasiya Rytova  and
Elena Yarovaya
Anastasiya Rytova
Affiliation:
Department of Probability Theory, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow119234, Russia (rytova.anastasiya@gmail.com; yarovaya@mech.math.msu.su)
Elena Yarovaya
Affiliation:
Department of Probability Theory, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow119234, Russia (rytova.anastasiya@gmail.com; yarovaya@mech.math.msu.su)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a continuous-time branching random walk (BRW) on the lattice ℤd, d ∈ ℕ, with a single source of branching, that is the lattice point where the birth and death of particles can occur. The random walk is assumed to be spatially homogeneous, symmetric and irreducible but, in contrast to the majority of previous investigations, the random walk transition intensities a(x, y) decrease as |y − x|−(d+α) for |y − x| → ∞, where α ∈ (0, 2), that leads to an infinite variance of the random walk jumps. The mechanism of the birth and death of particles at the source is governed by a continuous-time Markov branching process. The source intensity is characterized by a certain parameter β. We calculate the long-time asymptotic behaviour for all integer moments for the number of particles at each lattice point and for the total population size. With respect to the parameter β, a non-trivial critical point βc > 0 is found for every d ≥ 1. In particular, if β > βc the evolutionary operator generated a behaviour of the first moment for the number of particles has a positive eigenvalue. The existence of a positive eigenvalue yields an exponential growth in t of the particle numbers in the case β > βc called supercritical. Classification of the BRW treated as subcritical (β < βc) or critical (β = βc) for the heavy-tailed random walk jumps is more complicated than for a random walk with a finite variance of jumps. We study the asymptotic behaviour of all integer moments of a number of particles at any point y ∈ ℤd and of the particle population on ℤd according to the ratio d/α.

Keywords

Branching random walkheavy tailscritical casesubcritical casemomentsmultidimensional lattices

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 3 , June 2021 , pp. 971 - 992
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agbor, A., Molchanov, S. and Vainberg, B.. Global limit theorems on the convergence of multidimensional random walks to stable processes. Stoch. Dyn. 15 (2015), 1550024, 14.CrossRefGoogle Scholar
Albeverio, S., Bogachev, L. and Yarovaya, E.. Asymptotics of branching symmetric random walk on the lattice with a single source. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 975–980.CrossRefGoogle Scholar
Albeverio, S., Bogachev, L. V. and Yarovaya, E. B.. Branching random walk with a single source. In Communications in difference equations (Poznan, 1998), (eds. Elaydi, S., Ladas, G., Popenda, J. and Rakowski, J.) Amsterdam: Gordon and Breach, (2000), 9–19.Google Scholar
Albeverio, S., Bogachev, L., Molchanov, S. and Yarovaya, E.. Annealed moment Lyapunov exponents for a branching random walk in a homogeneous random branching environment. Markov Process. Related Fields. 6 (2000), 473–516.Google Scholar
Borovkov, A.and Borovkov, K.. Asymptotic Analysis of Random Walks Heavy-Tailed Distributions. (Cambridge: Cambridge University Press, 2008).CrossRefGoogle Scholar
Bulinskaya, E. Vl.. Complete classification of catalytic branching processes. Theory Probab. Appl. 59 (2015), 545–566.CrossRefGoogle Scholar
Dawson, D. A., Gorostiza, L. G. and Wakolbinger, A.. Occupation time fluctuations in branching systems. J. Theor. Probab. 14 (2001), 729–796.CrossRefGoogle Scholar
Feller, W.. An introduction to probability theory and its applications, vol. II, 2nd ed. (New York-London-Sydney: John Wiley & Sons Inc., 1971).Google Scholar
Gao, Z.-Q.. Exact convergence rate of the local limit theorem for branching random walks on the integer lattice. Stoch. Process. Appl. 127 (2017), 1282–1296.CrossRefGoogle Scholar
Getan, A., Molchanov, S. and Vainberg, B.. Intermittency for branching walks with heavy tails. Stoch. Dyn. 17 (2017), 1750044, 14.CrossRefGoogle Scholar
Gihman, I. I., Skorohod, A. V.. The theory of stochastic processes II. (Grundlehren Math. Wiss., 218, New York-Heidelberg: Springer-Verlag, (1975).Google Scholar
Gorostiza, L. G. and Wakolbinger, A.. Persistence criteria for a class of critical branching particle systems in continuous time. Ann. Probab. 19 (1991), 266–288.CrossRefGoogle Scholar
Khristolyubov, I. and Yarovaya, E. B.. A limit theorem for supercritical branching random walks with branching sources of varying intensity. Theory Probab. Appl. 64 (2019), 365–384.CrossRefGoogle Scholar
Kondratiev, Y., Molchanov, S., Pirogov, S. and Zhizhina, E.. On ground state of some non local Schrödinger operators. Appl. Anal. 96 (2016), 1390–1400.CrossRefGoogle Scholar
Kozyakin, V.. Hardy type asymptotics for cosine series in several variables with decreasing power-like coefficients. Int. J. Adv. Res. Math. 5 (2016), 35–51.CrossRefGoogle Scholar
Platonova, M. V. and Ryadovkin, K. S.. Branching random walks on ℤd with periodically distributed branching sources. Theory Probab. Appl. 64 (2019), 229–248.CrossRefGoogle Scholar
Rytova, A. I.. Harmonic analysis of branching random walks with heavy tails. Fundam. Prikl. Mat. 23:2 (2020), 150–164.Google Scholar
Rytova, A. I. and Yarovaya, E. B.. Multidimensional Watson lemma and its applications. Math. Notes. 99 (2016), 406–412.CrossRefGoogle Scholar
Rytova, A. and Yarovaya, E.. Survival analysis of particle populations in branching random walks. Commun. Statist. Simul. Comput. 99 (2019), 1–16.CrossRefGoogle Scholar
Rytova, A. I. and Yarovaya, E. B.. Moments of the numbers of particles in a heavy-tailed branching random walk. Uspekhi Mat. Nauk. 74 (2019), 165–166.Google Scholar
Sevastyanov, B. A.. Branching stochastic processes for particles diffusing in a bounded domain with absorbing boundaries. Teor. Veroyatnost. Primenen, 3 (1958), 121–136.Google Scholar
Shi, Z.. Branching random walks. Lecture notes from the 42nd Probability Summer School held in Saint Flour, 2012, Lecture Notes in Mathematics 2151. École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School]. (Cham: Springer, 2015).Google Scholar
Vatutin, V. A. and Topchii, V. A.. Catalytic branching random walks in ℤd with branching at the origin. Siberian Adv. Math. 23 (2013), 125–153.Google Scholar
Yarovaya, E. B.. Application of spectral methods in the study of branching processes with diffusion in a noncompact phase space. Theor. Math. Phys. 88 (1991), 690–694.CrossRefGoogle Scholar
Yarovaya, E. B.. Branching random walks in a heterogeneous environment (in Russian), (Moscow: Centre of Applied Investigations of the Faculty of Mechanics and Mathematics of the Moscow State University, 2007).Google Scholar
Yarovaya, E. B.. Models of branching walks and their use in reliability theory. Autom. Remote Control. 71 (2010), 1308–1324.CrossRefGoogle Scholar
Yarovaya, E.. Branching random walks with heavy tails. Commun. Statist. Theory Methods (16) 42 (2013), 2301–2310.Google Scholar
Yarovaya, E. B.. Branching random walks with several sources. Math. Popul. Stud. 20 (2013), 14–26.CrossRefGoogle Scholar
Yarovaya, E. B.. Spectral asymptotics of a supercritical branching random walk. Theory Probab. Appl. 62 (2018), 413–431.CrossRefGoogle Scholar
Zhizhina, E., Komech, S. and Descombes, X.. Modelling axon growing using CTRW. (2015), arXiv preprint arXiv:1512.02603.Google Scholar