scholarly journals Characterization of Critical Values of Branching Random Walks on Weighted Graphs through Infinite-Type Branching Processes

2008 ◽  
Vol 134 (1) ◽  
pp. 53-65 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walks ◽  
Branching Processes ◽  
Critical Values ◽  
Weighted Graphs ◽  
Infinite Type ◽  
Branching Random Walks

scholarly journals Critical Behaviors and Critical Values of Branching Random Walks on Multigraphs

2008 ◽  
Vol 45 (02) ◽  
pp. 481-497 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walks ◽  
Large Class ◽  
Geometric Parameter ◽  
Critical Value ◽  
Critical Values ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Bounded Degree ◽  
Branching Random Walks ◽  
Weak Phase

We consider weak and strong survival for branching random walks on multigraphs with bounded degree. We prove that, at the strong critical value, the process dies out locally almost surely. We relate the weak critical value to a geometric parameter of the multigraph. For a large class of multigraphs (which enlarges the class of quasi-transitive or regular graphs), we prove that, at the weak critical value, the process dies out globally almost surely. Moreover, for the same class, we prove that the existence of a pure weak phase is equivalent to nonamenability. The results are extended to branching random walks on weighted graphs.

scholarly journals Critical Behaviors and Critical Values of Branching Random Walks on Multigraphs

2008 ◽  
Vol 45 (2) ◽  
pp. 481-497 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walks ◽  
Large Class ◽  
Geometric Parameter ◽  
Critical Value ◽  
Critical Values ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Bounded Degree ◽  
Branching Random Walks ◽  
Weak Phase

We consider weak and strong survival for branching random walks on multigraphs with bounded degree. We prove that, at the strong critical value, the process dies out locally almost surely. We relate the weak critical value to a geometric parameter of the multigraph. For a large class of multigraphs (which enlarges the class of quasi-transitive or regular graphs), we prove that, at the weak critical value, the process dies out globally almost surely. Moreover, for the same class, we prove that the existence of a pure weak phase is equivalent to nonamenability. The results are extended to branching random walks on weighted graphs.

scholarly journals An Almost-Sure Renewal Theorem for Branching Random Walks on the Line

2010 ◽  
Vol 47 (03) ◽  
pp. 811-825
Author(s):  
Matthias Meiners
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
General Characteristic ◽  
Renewal Theorem ◽  
The Real ◽  
Branching Random Walks

In the present paper an almost-sure renewal theorem for branching random walks (BRWs) on the real line is formulated and established. The theorem constitutes a generalization of Nerman's theorem on the almost-sure convergence of Malthus normed supercritical Crump-Mode-Jagers branching processes counted with general characteristic and Gatouras' almost-sure renewal theorem for BRWs on a lattice.

scholarly journals An Almost-Sure Renewal Theorem for Branching Random Walks on the Line

2010 ◽  
Vol 47 (3) ◽  
pp. 811-825 ◽  
Author(s):  
Matthias Meiners
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
General Characteristic ◽  
Renewal Theorem ◽  
The Real ◽  
Branching Random Walks

In the present paper an almost-sure renewal theorem for branching random walks (BRWs) on the real line is formulated and established. The theorem constitutes a generalization of Nerman's theorem on the almost-sure convergence of Malthus normed supercritical Crump-Mode-Jagers branching processes counted with general characteristic and Gatouras' almost-sure renewal theorem for BRWs on a lattice.

scholarly journals A Note on the Transience of Critical Branching Random Walks on the Line

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Gerold Alsmeyer ◽  
Matthias Meiners
Keyword(s):  
Asymptotic Behavior ◽  
Random Walks ◽  
Lyapunov Functions ◽  
General Framework ◽  
Final Section ◽  
Branching Processes ◽  
Branching Random Walk ◽  
Branching Random Walks ◽  
International Audience ◽  
Critical Branching Random Walk

International audience Gantert and Müller (2006) proved that a critical branching random walk (BRW) on the integer lattice is transient by analyzing this problem within the more general framework of branching Markov chains and making use of Lyapunov functions. The main purpose of this note is to show how the same result can be derived quite elegantly and even extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of certain associated random weighted location measures which, upon taking expectations, provide a useful connection to the well established theory of ordinary random walks with i.i.d. increments. A brief discussion of the asymptotic behavior of the left- and rightmost particles in a critical BRW as time goes to infinity is provided in the final section by drawing on recent work by Hu and Shi (2008).

scholarly journals Harry Kesten’s work in probability theory

2021 ◽  
Author(s):  
Geoffrey R. Grimmett
Keyword(s):  
Probability Theory ◽  
Random Walks ◽  
Branching Processes

AbstractWe survey the published work of Harry Kesten in probability theory, with emphasis on his contributions to random walks, branching processes, percolation, and related topics.

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.

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.

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