A general continuous-state nonlinear branching process

Li, Pei-Sen; Yang, Xu; Zhou, Xiaowen

doi:10.1214/18-aap1459

Cited by 38 publications

(84 citation statements)

References 52 publications

(53 reference statements)

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“…It is also worth noticing that in the scaling limits, the nonlinearity or the environment can impact the diffusion or jump terms, and not only the drift as for BPILE considered in Section 4. One could also prove limits to CSBP with Lévy environment, where the jump measure associated with the demographical stochasticity (large jumps coming from the offsprings of one single individual, at a rate proportional to the number of individuals) is impacted by the environment, see [4], [29] for an example. Note also that we observe that one may want to go beyond the boundedness assumptions on the characteristics G N .…”

Section: Perspectives and Multidimensional Population Modelsmentioning

confidence: 99%

“…Lamperti has also introduced a powerful transform in the stable framework, see e.g. [25] and [29] and [5]. Other time changes have been successfully used to obtain scaling limits of discrete processes, in particular for some diffusion approximations, see for instance [21] for branching processes in random environment, [9] for branching processes with immigration and [33] for controlled branching processes , amongst others.…”

Section: Introductionmentioning

confidence: 99%

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Scaling limits of population and evolution processes in random environment

Bansaye¹,

Caballero²

2019

Electron. J. Probab.

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Our motivation comes from the large population approximation of individual based models in population dynamics and population genetics. We propose a general method to investigate scaling limits of finite dimensional Markov chains to diffusions with jumps. The results of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the chain transition. They strongly exploit the structure of the population processes recursively defined as sums of independent random variables. We develop two main applications. First, we extend the classical Wright-Fisher diffusion approximation to independent and identically distributed random environments. Second, we obtain the convergence in law of generalized Galton-Watson processes with interaction in random environment to the solution of stochastic differential equations with jumps.

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Section: Perspectives and Multidimensional Population Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Scaling limits of population and evolution processes in random environment

Bansaye¹,

Caballero²

2019

Electron. J. Probab.

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“…其中 {B t : t 0} 是 (G t ) Brown 运动. 我们在文献 [13] 中已经证明, 在上面条件成立的情形下, (1.3) 和 (1.4) 有唯一的非负强解. 文献 [13] 讨论了这一类过程在有限时间内灭绝、爆炸以及无穷远点为流入边界的条件.…”

Section: )unclassified

“…例如, 在某些应用模型中, 当粒子个体总数很大或者粒子处于高速运动中时, 不同个体间可能存在相互影响, 这导致生死速率可能会提高或降低, 且不能再由线性函数给出. 为了解决上述问题, 我们在文献 [13] 中考虑了连续状态非线性分枝过程. 直观上, 该过程是一类分枝速率依赖于当前人口的分枝过程.…”

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The discrete approximation of a class of continuous-state nonlinear branching processes

Li¹,

Xu²,

Zhou

2019

Sci. Sin.-Math.

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In this paper we consider a general continuous-state nonlinear branching process which can be identified as a nonnegative solution to a nonlinear version of the stochastic differential equation driven by Brownian motion and Poisson random measure. Intuitively, this process is a branching process with population-size-dependent branching rates and with competition. We construct a sequence of discrete-state nonlinear branching processes and prove that it converges weakly to the continuous-state nonlinear branching process by using tightness arguments and convergence criteria on infinite-dimensional space.

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“…Then (L, D) generates a CB-process with competition; see, e.g., Berestycki et al (2018), Lambert (2005) and Pardoux (2016). A more general class of population models, called continuousstate nonlinear branching processes, have been studied in Li (2018a) and Li et al (2017+).…”

Section: Introductionmentioning

confidence: 99%

Sample paths of continuous-state branching processes with dependent immigration

2019

Stochastic Models

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We prove the existence and pathwise uniqueness of the solution to a stochastic integral equation driven by Poisson random measures based on Kuznetsov measures for a continuous-state branching process. That gives a direct construction of the sample path of a continuous-state branching process with dependent immigration. The immigration rates depend on the population size via some functions satisfying a Yamada-Watanabe type condition. We only assume the existence of the first moment of the process. The existence of excursion law for the continuous-state branching process is not required. By special choices of the ingredients, we can make changes in the branching mechanism or construct models with competition.

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A general continuous-state nonlinear branching process

Abstract: In this paper we consider the unique nonnegative solution to the following generalized version of the stochastic differential equation for a continuous-state branching process.

Cited by 38 publications

References 52 publications

Scaling limits of population and evolution processes in random environment

Scaling limits of population and evolution processes in random environment

The discrete approximation of a class of continuous-state nonlinear branching processes

Sample paths of continuous-state branching processes with dependent immigration

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