Lina Ji, Zenghu Li
We give constructions of age-structured branching processes without or with immigration as pathwise unique solutions to stochastic integral equations. A necessary and sufficient condition for the ergodicity of the model with immigration is also given.
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Remaining-lifetime age-structured branching processes
本文研究了繁殖律依赖于亲本剩余寿命的年龄结构分支模型，通过随机方程构造了无移民和有移民的测度值过程，并在次临界情形下证明了遍历性条件及逗留时间的极限定理。
Limit theorems for supercritical remaining-lifetime age-structured branching processes
本文研究超临界年龄结构分支模型，给出马尔萨斯归一化随机测度收敛的充要条件，并在特定条件下加强极限理论，证明随机归一化的中心极限定理。
We study the long time behavior of an asymmetric size-structured measure-valued growth-fragmentation branching process that models the dynamics of a population of cells taking into account physiological and morphological asymmetry at division. We show that the process exhibits a Malthusian behavior; that is that the global population size grows exponentially fast and that the trait distribution of individuals converges to some stable distribution. The proof is based on a generalization of Lyapunov function techniques for non-conservative semi-groups. We then investigate the fluctuations of the…
We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of the descendants at birth depends on the trait of the mother and on the number of descendants. In this article, we explicitly describe the penalized Markov process, named auxiliary process, corresponding to the dynamic of the trait along the spine by giving its associated infinitesimal generator. We prove a Many-to-One formula and a Many-to-O…
This paper studies: (i) the long time behaviour of the empirical distribution of age and normalised position of an age dependent critical branching Markov process conditioned on non-extinction; and (ii) the super-process limit of a sequence of age dependent critical branching Brownian motions.
A new formulation of non-local branching superprocesses is given from which we derive as special cases the rebirth, the multitype, the mass-structured, the multilevel and the age-reproduction-structured superprocesses and the superprocess-controlled immigration process. This unified treatment simplifies considerably the proof of existence of the old classes of superprocesses and also gives rise to some new ones.
Structured coalescents, coagulation equations and multi-type branching processes
本文研究了结构人口中祖先谱系的凝聚过程，通过分析临界和大样本情形，发现经验测度过程收敛于d维凝聚方程的解，分别由多型分支过程和进入法表示。
The occupation time of an age-dependent branching particle system in $\Rd$ is considered, where the initial population is a Poisson random field and the particles are subject to symmetric $α$-stable migration, critical binary branching and random lifetimes. Two regimes of lifetime distributions are considered: lifetimes with finite mean and lifetimes belonging to the normal domain of attraction of a $γ$-stable law, $γ\in(0,1)$. It is shown that in dimensions $d>αγ$ for the heavy-tailed lifetimes case, and $d>α$ for finite mean lifetimes, the occupation time proccess satisfies a strong law of l…
We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of a descendant at birth depends on the trait of the mother. We prove a law of large numbers for the empirical distribution of ancestral trajectories. It ensures that the empirical measure converges to the mean value of the spine which is a time-inhomogeneous Markov process describing the trait of a typical individual along its ancestral lineag…
Long-time behaviour of a multidimensional age-dependent branching process with a singular jump kernel modelling telomere shortening
本文研究了具有奇异跳跃核的多维年龄依赖分支过程的长期行为，通过随机比较和弱Harnack不等式方法，证明了指数平稳性，并验证了主要结论的假设。
We study the exponential growth of bifurcating processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new cells's numbers are also assumed to be dependent. We illustrate our results by examples, including some Markov models. Our approach is related to the behaviour of the Laplace transform of nonnegative additive functional of Markov chains and require weak moment assumption (no exponential moment is needed).
Spinal constructions for continuous type-space branching processes with interactions
针对连续时间下结构化相互作用种群的分支过程问题，提出基于脊构造的Girsanov型结果与多对一公式，推导含相互作用的广义连续时间Kesten-Stigum定理，还给出用于规模依赖随机种群精确模拟的脊构造替代方法。
Branching random walks with ageing
本研究探讨了衰老对分支过程行为的影响，分析了参数修改对过程命运的影响。
We study the distribution of ages in the mean field forest fire model introduced by Ráth and Tóth. This model is an evolving random graph whose dynamics combine Erdős-Rényi edge-addition with a Poisson rain of lightning strikes. All edges in a connected component are deleted when any of its vertices is struck by lightning. We consider the asymptotic regime of lightning rates for which the model displays self-organized criticality. The age of a vertex increases at unit rate, but it is reset to zero at each burning time. We show that the empirical age distribution converges as a process to a det…
We establish convergence to the Kingman coalescent for a class of age-structured population models with time-constant population size. Time is discrete with unit called a year. Offspring numbers in a year may depend on mother's age.
Occupation-Time Fluctuations of an Age-Dependent Branching System Driven by Poisson Immigration
研究临界年龄依赖分支粒子系统在泊松移民驱动下的占据时间波动，证明在适当时空尺度下，重标度波动弱收敛到具有明确协方差结构的高斯过程，该过程自相似且具有长程依赖性。
It is well known that a simple, supercritical Bienaymé-Galton-Watson process turns into a subcritical such process, if conditioned to die out. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction.
We study explosiveness of age-dependent branching processes describing the early stages of an epidemic-spread: both forward- and backward process are analysed. For the classical age-dependent branching process $(h,G)$, where the offspring has probability generating function $h$ and all individuals have life-lengths independently picked from a distribution $G$, we focus on the setting $h = h_α^L$, with $L$ a function varying slowly at infinity and $α\in (0,1)$. Here, $h^L_α(s) = 1 - (1-s)^α L(\frac{1}{1-s}),$ as $s \to 1$. For a fixed $G$, the process $(h^L_α,G)$ explodes either for all $α\in (…
Occupation time fluctuations of an age-dependent critical binary branching particle system
本文研究年龄依赖临界二元分支粒子系统占据时间过程的极限波动，针对有限期望和帕累托型寿命分布，证明了弱收敛于加权子分数布朗运动，并探讨其性质与参数范围。
Non-linear branching processes and Crump-Mode-Jagers processes with interaction
本文研究带交互的Crump-Mode-Jagers过程，证明其局部拓扑下树结构的大数定律，形式化典型个体树独立性的普适原理，提出非线性分支过程刻画局部树结构分布。
A decomposable strongly critical Galton-Watson branching process with $N$ types of particles labelled $1,2,...,N$ is considered in which a type~$i$ parent may produce individuals of types $j\geq i$ only. This model may be viewed as a stochastic model for the sizes of a geographically structured population occupying $N$ islands, the location of a particle being considered as its type. The newborn particles of island $i\leq N-1$ either stay at the same island or migrate, just after their birth to the islands $% i+1,i+2,...,N$. Particles of island $N$ do not migrate. We investigate the structure …
We study multi-type Bienaymé-Galton-Watson processes with linear-fractional reproduction laws using various analytical tools like contour process, spinal representation, Perron-Frobenius theorem for countable matrices, renewal theory. For this special class of branching processes with countably many types we present a transparent criterion for $R$-positive recurrence with respect to the type space. This criterion appeals to the Malthusian parameter and the mean age at childbearing of the associated linear-fractional Crump-Mode-Jagers process.
We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate theta, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have i.i.d. lifetimes durations (non necessarily exponential) during which they give birth independently at constant rate b. First, using spine decomposition, we relax previously known assumptions required for a.s. convergence of total population size. Then, we consider three models of structured populations: …
A superprocess limit for an interacting birth-death particle system modelling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while ages are set to zero. Because of interactions between individuals, standard approaches based on the Laplace transform do not hold. We use a martingale problem approach and a separation of the slow (trait) and fast (age) scales. While the trait marginals converge in a pathwise sense to a superprocess, the age distributions, on another time scale, average to e…
Age dependent population dynamics are frequently modeled with generalizations of the classic McKendrick-von Foerster equation. These are deterministic systems, and a stochastic generalization was recently reported in [1,2]. Here we develop a fully stochastic theory for age-structured populations via quantum field theoretical Doi-Peliti techniques. This results in a path integral formulation where birth and death events correspond to cubic and quadratic interaction terms. This formalism allows us to efficiently recapitulate the results in [1,2], exemplifying the utility of Doi-Peliti methods. F…
Branching processes and bacterial growth
本文提出一个分段确定性的马尔可夫分支树模型，用于刻画具有年轻极与年老极两类细菌在大小依赖分裂率及个体化增长率下的群体生长，并证明其良定义性、多对一公式及平均经验测度满足增长-碎裂方程。
Two-type branching processes with immigration, and the structured coalescents
本文研究了两种类型个体在两个岛屿上的分支过程，通过两型Feller扩散模型探讨了频率变化，并推导了不对称两岛频率过程的性质及大种群极限，揭示了平衡选择的新机制。
Age-structured stochastic populations under dynamic harvesters' behavior: well-posedness, asymptotic stability and numerically-amenable approximations
本文提出年龄结构种群与收获者反馈的个体模型，证明收敛到耦合随机微分方程，并建立适定性、正性、灭绝边界不可达性及长期行为，附数值实验验证。
In this paper we initiate the theory of Crump-Mode-Jagers branching processes (BP) in the setting where no Malthusian parameter exist, i.e., the process grows faster than exponential. A Crump-Mode-Jagers BP is a branching process (in continuous time) where arbitrary dependencies are allowed between the birth-times of the children of a single individual in the population. It is however assumed that these reproduction processes are i.i.d. point processes for different individuals. This paper focuses on determining whether this branching process explodes, that is, the process reaches infinitely m…
Parameter inference for partially observed branching processes
研究年龄依赖分支过程，目标是仅依据种群总数观测估计模型参数，核心方法是引入矩估计法并建立渐近正态性，主要贡献是实现参数推断并讨论了模型扩展。
We study the nonparametric estimation of the branching rate $B(x)$ of a supercritical Bellman-Harris population: a particle with age $x$ has a random lifetime governed by $B(x)$; at its death time, it gives rise to $k \geq 2$ children with lifetimes governed by the same division rate and so on. We observe in continuous time the process over $[0,T]$. Asymptotics are taken as $T \rightarrow \infty$; the data are stochastically dependent and one has to face simultaneously censoring, bias selection and non-ancillarity of the number of observations. In this setting, under appropriate ergodicity pro…
We study mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we present a full kinetic framework for age-structured interacting populations undergoing birth, death and fission processes, in spatially dependent environments. We define the complete probability density for the population-size-age-chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalizati…
We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. …
Mean field games are concerned with the limit of large-population stochastic differential games where the agents interact through their empirical distribution. In the classical setting, the number of players is large but fixed throughout the game. However, in various applications, such as population dynamics or economic growth, the number of players can vary across time which may lead to different Nash equilibria. For this reason, we introduce a branching mechanism in the population of agents and obtain a variation on the mean field game problem. As a first step, we study a simple model using …
We prove the well-posedness of a differential equation that describes the evolution of the large-system limit of the empirical age measure in the mean field forest fire model of Ráth and Tóth (arXiv:0808.2116). This forest fire model is a random graph process on $n$ vertices, whose dynamics combine the Erdős-Rényi dynamics with a Poisson rain of lightning strikes. All edges in any connected component are deleted as soon as any of its vertices is struck by lightning. Each vertex has an age, which increases at rate $1$ but is reset to $0$ each time it burns. We consider the asymptotic lightning …
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