scholarly journals The probabilities of extinction in a branching random walk on a strip

2020 ◽  
Vol 57 (3) ◽  
pp. 811-831
Author(s):  
Peter Braunsteins ◽  
Sophie Hautphenne
Keyword(s):  
Random Walk ◽  
Iterative Method ◽  
Fixed Points ◽  
Branching Processes ◽  
Extinction Probability ◽  
Branching Random Walk ◽  
Infinite Type ◽  
Hessenberg Form ◽  
Global Extinction ◽  
Countably Infinite

AbstractWe consider a class of multitype Galton–Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction probability $\tilde{\textbf{\textit{q}}}$ , that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for $\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$ . We then present an iterative method to compute the vector $\textbf{\textit{q}}(A)$ for any set A. Finally, we investigate the location of the vectors $\textbf{\textit{q}}(A)$ in the set of fixed points of the progeny generating vector.

Fixed points of a generalized smoothing transformation and applications to the branching random walk

1998 ◽  
Vol 30 (01) ◽  
pp. 85-112 ◽  
Author(s):  
Quansheng Liu
Keyword(s):  
Fixed Point ◽  
Random Walk ◽  
Fixed Points ◽  
Branching Processes ◽  
Random Variables ◽  
Initial Distribution ◽  
Branching Random Walk ◽  
Independent Random Variables ◽  
Sorting Algorithm ◽  
Hausdorff Measures

Let {Ai:i≥ 1} be a sequence of non-negative random variables and letMbe the class of all probability measures on [0,∞]. Define a transformationTonMby lettingTμ be the distribution of ∑i=1∞AiZi, where theZiare independent random variables with distribution μ, which are also independent of {Ai}. Under first moment assumptions imposed on {Ai}, we determine exactly whenThas a non-trivial fixed point (of finite or infinite mean) and we prove that all fixed points have regular variation properties; under moment assumptions of order 1 + ε, ε &gt; 0, we findallthe fixed points and we prove that all non-trivial fixed points have stable-like tails. Convergence theorems are given to ensure that each non-trivial fixed point can be obtained as a limit of iterations (byT) with an appropriate initial distribution; convergence to the trivial fixed points δ0and δ∞is also examined, and a result like the Kesten-Stigum theorem is established in the case where the initial distribution has the same tails as a stable law. The problem of convergence with an arbitrary initial distribution is also considered when there is no non-trivial fixed point. Our investigation has applications in the study of: (a) branching processes; (b) invariant measures of some infinite particle systems; (c) the model for turbulence of Yaglom and Mandelbrot; (d) flows in networks and Hausdorff measures in random constructions; and (e) the sorting algorithm Quicksort. In particular, it turns out that the basic functional equation in the branching random walk always has a non-trivial solution.

Fixed points of a generalized smoothing transformation and applications to the branching random walk

1998 ◽  
Vol 30 (1) ◽  
pp. 85-112 ◽  
Author(s):  
Quansheng Liu
Keyword(s):  
Fixed Point ◽  
Random Walk ◽  
Fixed Points ◽  
Branching Processes ◽  
Random Variables ◽  
Initial Distribution ◽  
Branching Random Walk ◽  
Independent Random Variables ◽  
Sorting Algorithm ◽  
Hausdorff Measures

Let {Ai : i ≥ 1} be a sequence of non-negative random variables and let M be the class of all probability measures on [0,∞]. Define a transformation T on M by letting Tμ be the distribution of ∑i=1∞AiZi, where the Zi are independent random variables with distribution μ, which are also independent of {Ai}. Under first moment assumptions imposed on {Ai}, we determine exactly when T has a non-trivial fixed point (of finite or infinite mean) and we prove that all fixed points have regular variation properties; under moment assumptions of order 1 + ε, ε > 0, we find all the fixed points and we prove that all non-trivial fixed points have stable-like tails. Convergence theorems are given to ensure that each non-trivial fixed point can be obtained as a limit of iterations (by T) with an appropriate initial distribution; convergence to the trivial fixed points δ0 and δ∞ is also examined, and a result like the Kesten-Stigum theorem is established in the case where the initial distribution has the same tails as a stable law. The problem of convergence with an arbitrary initial distribution is also considered when there is no non-trivial fixed point. Our investigation has applications in the study of: (a) branching processes; (b) invariant measures of some infinite particle systems; (c) the model for turbulence of Yaglom and Mandelbrot; (d) flows in networks and Hausdorff measures in random constructions; and (e) the sorting algorithm Quicksort. In particular, it turns out that the basic functional equation in the branching random walk always has a non-trivial solution.

scholarly journals Volume explored by a branching random walk on general graphs

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Ignacio Bordeu ◽  
Saoirse Amarteifio ◽  
Rosalba Garcia-Millan ◽  
Benjamin Walter ◽  
Nanxin Wei ◽  
...  
Keyword(s):  
Random Walk ◽  
Metabolic Networks ◽  
Branching Processes ◽  
Nuclear Fission ◽  
Random Trees ◽  
Branching Random Walk ◽  
Scale Free ◽  
Animal Populations ◽  
Wide Range ◽  
General Graphs

Abstract Branching processes are used to model diverse social and physical scenarios, from extinction of family names to nuclear fission. However, for a better description of natural phenomena, such as viral epidemics in cellular tissues, animal populations and social networks, a spatial embedding—the branching random walk (BRW)—is required. Despite its wide range of applications, the properties of the volume explored by the BRW so far remained elusive, with exact results limited to one dimension. Here we present analytical results, supported by numerical simulations, on the scaling of the volume explored by a BRW in the critical regime, the onset of epidemics, in general environments. Our results characterise the spreading dynamics on regular lattices and general graphs, such as fractals, random trees and scale-free networks, revealing the direct relation between the graphs’ dimensionality and the rate of propagation of the viral process. Furthermore, we use the BRW to determine the spectral properties of real social and metabolic networks, where we observe that a lack of information of the network structure can lead to differences in the observed behaviour of the spreading process. Our results provide observables of broad interest for the characterisation of real world lattices, tissues, and networks.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

Tightness for the minimal displacement of branching random walk

2007 ◽  
Vol 2007 (07) ◽  
pp. P07010-P07010 ◽  
Author(s):  
Maury Bramson ◽  
Ofer Zeitouni
Keyword(s):  
Random Walk ◽  
Branching Random Walk

Two-Dimensional Limit Theorem for a Critical Catalytic Branching Random Walk

2004 ◽  
pp. 387-395 ◽  
Author(s):  
Valentin Topchii ◽  
Vladimir Vatutin
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Branching Random Walk ◽  
Two Dimensional ◽  
Catalytic Branching

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (2) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating function G and a maximum principle which, we prove, is satisfied by every fixed point of G. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

Spectral Asymptotics of a Supercritical Branching Random Walk

2018 ◽  
Vol 62 (3) ◽  
pp. 413-431 ◽  
Author(s):  
E. B. Yarovaya
Keyword(s):  
Random Walk ◽  
Spectral Asymptotics ◽  
Branching Random Walk

scholarly journals Branching random walk with selection at critical rate

Bernoulli ◽  
2017 ◽  
Vol 23 (3) ◽  
pp. 1784-1821
Author(s):  
Bastien Mallein
Keyword(s):  
Random Walk ◽  
Branching Random Walk ◽  
Critical Rate
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