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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 ([email protected]; [email protected])
Elena Yarovaya
Affiliation:
Department of Probability Theory, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow119234, Russia ([email protected]; [email protected])
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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
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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