Moments of particle numbers in a branching random walk with heavy tails

Rytova, A. I.; Yarovaya, Elena

doi:10.1070/rm9914

Cited by 3 publications

(7 citation statements)

References 5 publications

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“…Random walks with infinite variance of jumps are commonly referred to in the literature as random walks with heavy tails. We will consider the simplest case, when H(z/|z|) ≡ C > 0, and use the results obtained in [Rytova, Yarovaya (2019), Rytova, Yarovaya (2020)], where a BRW with one particle generation center and the absence of absorbing sources was considered under condition (3).…”

Section: Description Of the Modelmentioning

confidence: 99%

“…with the initial conditions q(0, •, y) = δ y (•) and q(0, •) ≡ 1 respectively. Note that the resulting equation has exactly the same form as the equation for the first moments in the BRW without absorbing sources, considered in [Yarovaya(2007)] (or in [Rytova, Yarovaya (2019)] for the case of heavy tails), which greatly simplifies the study. The classification of the asymptotic behavior of the first moments of the local number of particles and the total number of particles for arbitrary d−dimensional lattices in the considered BRW can be obtained using the classification of the asymptotic behavior for the functions q(t, x, y) and q(t, x), obtained in [Yarovaya(2007)], [Rytova, Yarovaya (2019)], and the relation m 1 = qe −b0t .…”

Section: Classification Of the Asymptotic Behavior Of The First Momentsmentioning

confidence: 99%

“…The behavior of a BRW with a single source of particle generation (branching) located at the origin and no absorption at other points under the assumption of a finite variance of jumps has been studied, for example, in [Yarovaya(2007)], and with infinite variance in [Rytova, Yarovaya (2019), Rytova, Yarovaya (2020)]. The random walk underlying the processes under consideration is defined using the transition intensity matrix A = (a(x, y)) x,y∈Z d and satisfies conditions of regularity, symmetry, spatial homogeneity (which allows us to consider a(x, y) as a function of one argument a(y − x)), homogeneity in time and irreducibility.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point

Filichkina¹,

Yarovaya²

2023

Preprint

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We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only absorption of particles can occur. The asymptotic behavior of the integer moments of both the total number of particles and the number of particles at a lattice point is studied depending on the relationship between the model parameters. In the case of the existence of an isolated positive eigenvalue of the evolution operator of the average number of particles, a limit theorem is obtained on the exponential growth of both the total number of particles and the number of particles at a lattice point.

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Section: Description Of the Modelmentioning

confidence: 99%

Section: Classification Of the Asymptotic Behavior Of The First Momentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point

Filichkina¹,

Yarovaya²

2023

Preprint

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“…First, we find the asymptotics of m 1 (t, x, 0). Substituting y = 0 into the integral equation (11), we obtain…”

Section: Subcritical Brwmentioning

confidence: 99%

“…For the case d/α ∈ (1, ∞), by the integral equation (11) we express m 1 (t, x, 0). We use the asymptotics of p(t, x, y) in ( 5) and the asymptotics of m 1 (t, 0, 0) in [11, theorem 3] to find the asymptotics of the integral by lemma 2 for confolutions from [16].…”

Section: Subcritical Brwmentioning

confidence: 99%

Heavy-Tailed Branching Random Walks on Multidimensional Lattices. A Moment Approach

Rytova¹,

Yarovaya²

2020

Preprint

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We study a continuous-time branching random walk on the lattice Z d , 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 homogeneous, symmetric and irreducible but, in contrast to 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 Bienaymé-Galton-Watson 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 branching random walk 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 ∈ Z d and of the particle population on Z d according to the ratio d/α.

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Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point

Filichkina

Yarovaya

2023

Mathematics

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We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only the absorption of particles can occur. The asymptotic behavior of the integer moments of both the total number of particles and the number of particles at a lattice point is studied depending on the relationship between the model parameters. In the case of the existence of an isolated positive eigenvalue of the evolution operator of the average number of particles, a limit theorem is obtained on the exponential growth of both the total number of particles and the number of particles at a lattice point.

show abstract

Moments of particle numbers in a branching random walk with heavy tails

Cited by 3 publications

References 5 publications

Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point

Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point

Heavy-Tailed Branching Random Walks on Multidimensional Lattices. A Moment Approach

Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point

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