Asymptotic properties of supercritical branching processes in random environments

Li, Yingqiu; Liu, Quansheng; Gao, Zhiqiang; Hesong, Wang

doi:10.1007/s11464-014-0397-z

Cited by 6 publications

(4 citation statements)

References 31 publications

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“…Branching processes in random environment (BPREs) were first introduced and studied in Smith and Wilkinson [37] and have attracted considerable interest in the last decade (e.g. [1,10,27] and the references therein). BPREs are interesting since they are more realistic models compared with classical branching processes and, from the mathematical point of view, they have new properties such as another phase transition in the subcritical regime.…”

Section: Introductionmentioning

confidence: 99%

Branching Processes in a Lévy Random Environment

Palau

Pardo

2017

Acta Appl Math

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In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813-857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1-13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition.

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Section: Introductionmentioning

confidence: 99%

Branching Processes in a Lévy Random Environment

Palau

Pardo

2017

Acta Appl Math

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“…The sufficient condition is only a little bit stronger (see Theorem 1,[25]). Under this sufficient condition, (13) and (14) hold.…”

Section: Further Resultsmentioning

confidence: 95%

“…Most of that was done for the i.i.d environment, because then properties of the so-called "associated random walks" could be applied, but some results hold also in a stationary and ergodic environment. For a sample of results see [3,5,4,8,9,13] and references therein.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Absolute continuity of the martingale limit in branching processes in random environment

Damek¹,

Gantert²,

Kolesko³

2019

Electron. Commun. Probab.

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We consider a supercritical branching process Z n in a stationary and ergodic random environment ξ = (ξ n ) n≥0 . Due to the martingale convergence theorem, it is known that the normalized population size W n = Z n /(E(Z n |ξ)) converges almost surely to a random variable W . We prove that if W is not concentrated at 0 or 1 then for almost every environment ξ the law of W conditioned on the environment ξ is absolutely continuous with a possible atom at 0. The result generalizes considerably the main result of [10], and of course it covers the well-known case of the martingale limit of a Galton-Watson process. Our proof combines analytical arguments with the recursive description of W .

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“…Branching processes in random environment (BPREs) were first introduced and studied in Smith and Wilkinson [33] and have attracted considerable interest in the last decade (e.g. [1,9,24,34] and the reference therein). BPREs are interesting since they are more realistic models compared with classical branching processes and, from the mathematical point of view, they have new properties such as phase transitions in the subcritical regime.…”

Section: Introductionmentioning

confidence: 99%

Branching processes in a Lévy random environment

Palau¹,

Pardo²

2015

Preprint

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In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by Brownian motions and Poisson random measures which are mutually independent. The existence and uniqueness of strong solutions are established under some general conditions that allows us to consider the case when the strong solution explodes at a finite time. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study some properties of such processes that extends recent results obtained by Bansaye et al. in (Electron.

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Asymptotic properties of supercritical branching processes in random environments

Cited by 6 publications

References 31 publications

Branching Processes in a Lévy Random Environment

Branching Processes in a Lévy Random Environment

Absolute continuity of the martingale limit in branching processes in random environment

Branching processes in a Lévy random environment

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