The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter λ. There is a threshold for λ, which is called λw, that separates almost sure global extinction from global survival. Analogously, there exists another threshold λs below which any site is visited almost surely a finite number of times (i.e. local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter λs is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter λw is the inverse of a certain function of the reproduction rates, which we denote by Kw. We provide here new sufficient conditions which guarantee that the global critical parameter equals 1/Kw. This result extends previously known results for branching random walks on multigraphs and general branching random walks. We show that these sufficient conditions are satisfied by periodic tree-like branching random walks. We also discuss the critical parameter and the critical behaviour of continuous-time branching processes in varying environment. So far, only examples where λw = 1/Kw were known; here we provide an example where λw > 1/Kw.

Bertacchi, D., Coletti, C., Zucca, F. (2017). Global survival of branching random walks and tree-like branching random walks. ALEA, 14(1), 381-402 [10.30757/alea.v14-21].

Global survival of branching random walks and tree-like branching random walks

Bertacchi, D;
2017

Abstract

The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter λ. There is a threshold for λ, which is called λw, that separates almost sure global extinction from global survival. Analogously, there exists another threshold λs below which any site is visited almost surely a finite number of times (i.e. local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter λs is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter λw is the inverse of a certain function of the reproduction rates, which we denote by Kw. We provide here new sufficient conditions which guarantee that the global critical parameter equals 1/Kw. This result extends previously known results for branching random walks on multigraphs and general branching random walks. We show that these sufficient conditions are satisfied by periodic tree-like branching random walks. We also discuss the critical parameter and the critical behaviour of continuous-time branching processes in varying environment. So far, only examples where λw = 1/Kw were known; here we provide an example where λw > 1/Kw.
Articolo in rivista - Articolo scientifico
Branching process; Branching random walk; Critical parameters; Generating function; Global survival; Local survival; Tree-like; Varying environment;
branching random walk, branching process, local survival, global survival, varying environment, tree-like, critical parameters, generating function
English
2017
14
1
381
402
reserved
Bertacchi, D., Coletti, C., Zucca, F. (2017). Global survival of branching random walks and tree-like branching random walks. ALEA, 14(1), 381-402 [10.30757/alea.v14-21].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/151208
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