Vyacheslav M. Abramov
Under the assumption that the initial population size of a Galton-Watson branching process increases to infinity, the paper studies asymptotic behavior of the population size before extinction. More specifically, we establish asymptotic properties of the conditional moments (which are exactly defined in the paper).
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In this note, we study the asymptotic behaviour near extinction of (sub-) critical continuous state branching processes. In particular, we establish an analogue of Khintchin's law of the iterated logarithm near extinction time for a continuous state branching process whose branching mechanism satisfies a given condition and its reflected process at its infimum.
Lower and upper bounds for the cumulative distribution function (cdf) of the time to extinction in a subcritical two-sex branching process are derived. A recursive procedure for approximating this cdf is also utilized. The results are illustrated with some simulations.
The asymptotic behavior, as $n\rightarrow \infty $ of the conditional distribution of the number of particles in a decomposable critical branching process $\mathbf{Z}% (m)=(Z_{1}(m),...,Z_{N}(m)),$ with $N$ types of particles at moment $m=n-k,\, k=o(n),$ is investigated given that the extinction moment of the process is $n$.
We give an explicit formula for the most likely path to extinction for the Galton-Watson processes with large initial population. We establish this result with the help of the large deviation principle (LDP) which also recovers the asymptotics of extinction probability. Due to the nonnegativity of the Galton-Watson processes, the proof of LDP verification at the point of extinction uses a nonstandard argument of independent interest.
Given a branching random walk on a set $X$, we study its extinction probability vectors $\mathbf q(\cdot,A)$. Their components are the probability that the process goes extinct in a fixed $A\subseteq X$, when starting from a vertex $x\in X$. The set of extinction probability vectors (obtained letting $A$ vary among all subsets of $X$) is a subset of the set of the fixed points of the generating function of the branching random walk. In particular here we are interested in the cardinality of the set of extinction probability vectors. We prove results which allow to understand whether the probab…
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 prove the exponential estimate \begin{equation*} P \{ s < τ< \infty \} \leq C e^{-q s}, \quad s \geq 0, \end{equation*} where $C, q >0$ are constants and $ τ$ is the extinction time of the supercritical branching random walk (BRW) on a cube. We cover both the discrete-space and continuous-space BRWs.
We consider a discrete-time host-parasite model for a population of cells which are colonized by proliferating parasites. The cell population grows like an ordinary Galton-Watson process, but in reflection of real biological settings the multiplication mechanisms of cells and parasites are allowed to obey some dependence structure. More precisely, the number of offspring produced by a mother cell determines the reproduction law of a parasite living in this cell and also the way the parasite offspring is shared into the daughter cells. In this article, we provide a formal introduction of this b…
Let $T$ be the extinction moment of a critical branching process $Z=(Z_{n},n\geq 0) $ in a random environment specified by iid probability generating functions. We study the asymptotic behavior of the probability of extinction of the process $Z$ at moment $n\to \infty$, and show that if the logarithm of the (random) expectation of the offspring number belongs to the domain of attraction of a non-gaussian stable law then the extinction occurs owing to very unfavorable environment forcing the process, having at moment $T-1$ exponentially large population, to die out. We also give an interpretati…
We study the speed of extinction of continuous state branching processes in a Lévy environment, where the associated Lévy process oscillates. Assuming that the Lévy process satisfies the Spitzer's condition and the existence of some exponential moments, we extend recent results where the associated branching mechanism was stable. Our study relies on the path analysis of the process together with its environment, when this latter is conditioned to have a non negative running infimum. This approach is inspired from the discrete setting with i.i.d. environment studied in (Afanasyev et al. 2005).
We consider the extinction events of Galton-Watson processes with countably infinitely many types. In particular, we construct truncated and augmented Galton-Watson processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the global extinction probability vector of the Galton-Watson processes with countably infinitely many types. This gives rise to a number of iterative methods for the computation of the global extinction probability vector.
We consider a class of multitype Galton-Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, the probability $\boldsymbol{q}(A)$ of extinction in subsets of types $A\subseteq \mathcal{X}_d$ may differ from the global extinction probability $\boldsymbol{q}$ and the partial extinction probability $\tilde{\boldsymbol{q}}$. After deriving partial and global extinction criteria, we develop conditions for $\boldsymbol{q}<\boldsymbol{q}(A)<\tilde{\boldsymbol{q}}$. We then present an iterative…
We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton-Watson processes with typeset $\mathcal{X}=\{0,1,2,\dots\}$, in which individuals of type $i$ may give birth to offspring of type $j\leq i+1$ only. For this class of processes, we study the set $S$ of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector $\boldsymbol{q}$ and whose maximum is the partial extinction pro…
In this paper, we consider time-inhomogeneous branching processes and time-inhomogeneous birth-and-death processes, in which the offspring distribution and birth and death rates (respectively) vary in time. A classical result of branching processes states that in the critical regime, a process conditioned on non-extinction and normalized will converge in distribution to a standard exponential. In a paper of Jagers, time-inhomogeneous branching processes are shown to exhibit this convergence as well. In this paper, the hypotheses of Jagers' result are relaxed, further hypotheses are presented f…
On Asymptotic Behavior of Extinction Moment of Critical Bisexual Branching Process in Random Environment
研究独立同分布随机环境下的临界双性分支过程，证明从大量配对 $N$ 开始时其灭绝时间阶为 $\ln^2 N$，且该结论适用于包括一夫一妻、一夫多妻等一般交配函数。
We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is …
A multitype continuous-state branching process (MCSBP) ${\rm Z}=({\rm Z}_{t})_{t\geq 0}$, is a Markov process with values in $[0,\infty)^{d}$ that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of $d$ Laplace exponents of $\mathbb{R}^d$-valued spectrally positive Lévy processes, each one having $d-1$ increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing…
This note gives an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to infinity). We only require a second moment for the offspring-type distribution featuring …
We consider continuous state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term extinction and explosion behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and, as in the case of branching processes in random environment, we find five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, we st…
Inverting the operation of conditioning a branching process on extinction
该研究针对分支过程的灭绝条件化操作提出逆问题，证明温和假设下共轭分支过程存在且多类型时不唯一，其结果可用于随机平面映射构造与癌症建模。
The asymptotic behavior, as $n\rightarrow \infty $ of the probability of the event that a decomposable critical branching process $\mathbf{Z}(m)=(Z_{1}(m),...,Z_{N}(m)),$ $m=0,1,2,...,$ with $N$ types of particles dies at moment $n$ is investigated and conditional limit theorems are proved describing the distribution of the number of particles in the process $\mathbf{Z}(\cdot)$ at moment $m<n,$ given that the extinction moment of the process is $n$. These limit theorems may be considered as the statements describing the distribution of the number of vertices in the layers of certain classes of…
We consider continuous state branching processes (CSBP) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a Lévy process with bounded variation paths. We construct a process of this class as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and to observe new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish the probability of …
In this manuscript, we continue with the systematic study of the speed of extinction of continuous state branching processes in Lévy environments under more general branching mechanisms. Here, we deal with the weakly subcritical regime under the assumption that the branching mechanism is regularly varying. We extend recent results of Li and Xu [14] and Palau et al. [17], where it is assumed that the branching mechanism is stable and complement the recent articles of Bansaye et al. [2] and by the authors in [7], where the critical and the strongly and intermediate subcritical cases were treated…
The purpose of this article is to observe that the zero sets of continuous-state branching processes with immigration (CBI) are infinitely divisible regenerative sets. Indeed, they can be constructed by the procedure of random cutouts introduced by Mandelbrot in 1972. We then show how very precise information about the zero sets of CBI can be obtained in terms of the branching and immigrating mechanism.
Extinction and extinguishment properties for a nonlinear predator-prey branching model
针对由布朗分支噪声和谱正稳定跳跃驱动的两类连续状态非线性捕食-被捕食分支模型，构建互补结构推导灭绝准则，发现有限时间灭绝与非灭绝共存 regime，明确非灭绝时种群联合收敛至零的性质。
Recently in Barczy, Li and Pap (2015), the notion of a multi-type continuous-state branching process (with immigration) having d-types was introduced as a solution to an d-dimensional vector- valued SDE. Preceding that, work on affine processes, originally motivated by math- ematical finance, in Duffie, Filipovic and Schachermayer (2003) also showed the existence of such processes. See also more recent contributions in this direction due to Gabrielli and Teichmann (2014) and Caballero, Perez Garmendia and Uribe Bravo (2015). Older work on multi-type continuous-state branching processes is more…
In this paper, we study the speed of extinction of continuous state branching processes in subcritical Lévy environments. More precisely, when the associated Lévy process to the environment drifts to $-\infty$ and, under a suitable exponential martingale change of measure (Esscher transform), the environment either drifts to $-\infty$ or oscillates. We extend recent results of Palau et al. (2016) and Li and Xu (2018), where the branching term is associated to a spectrally positive stable Lévy process and complement the recent article of Bansaye et al. (2021) where the critical case was studied…
The extinction rate of a branching random walk with a barrier in a time-inhomogeneous random environment
本文研究时间非齐次随机环境中带随机吸收屏障的超临界分支随机游走，给出屏障导致系统灭绝时在 $L^p$ 意义下的灭绝速率，并证明随机环境下随机游走小偏差概率的 $L^p$ 收敛性。
Markov branching processes with disasters: extinction, survival and duality to p-jump processes
本文通过建立 p-跳跃过程在单位区间上的极限定理，并利用概率生成函数对偶性，研究带二项灾难的分支过程的生存概率极限，进而推广到时间非齐次生灭过程和带 p-跳跃的连续状态分支过程。
Supercritical branching processes in constant environment conditioned on eventual extinction are known to be subcritical branching processes. The case of random environment is more subtle. A supercritical branching diffusion in random environment (BDRE) conditioned on eventual extinction of the population is not a BDRE. However the quenched law of the population size of a supercritical BDRE conditioned on eventual extinction is equal to the quenched law of the population size of a subcritical BDRE. As a consequence, supercritical BDREs have a phase transition which is similar to a well-known p…
We consider a critical branching particle system in $\R^d$, composed of individuals of a finite number of types $i\in\{1,...,K\}$. Each individual of type $i$ moves independently according to a symmetric $α_i$-stable motion. We assume that the particle lifetimes and offspring distributions are type-dependent. Under the usual independence assumptions in branching systems, we prove extinction theorems in the following cases: (1) all the particle lifetimes have finite mean, or (2) there is a type whose lifetime distribution has heavy tail, and the other lifetimes have finite mean. We get a more c…
We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly f…
We study a model of growing population that competes for resources. At each time step, all existing particles reproduce and the offspring randomly move to neighboring sites. Then at any site with more than one offspring, the particles are annihilated. This is a nonmonotone model, which makes the analysis more difficult. We consider the extinction window of this model in the finite mean-field case, where there are $n$ sites but movement is allowed to any site (the complete graph). We show that although the system survives for exponential time, the extinction window is logarithmic.
In this paper we study a 2-type linear-fractional branching process in varying environment with asymptotically constant mean matrices. Let $ν$ be the extinction time and for $k\ge1$ let $M_k$ be the mean matrix of offspring distribution of individuals of the $(k-1)$-th generation. Under certain conditions, we show that $P(ν=n)$ and $P(ν>n)$ are asymptotically equivalent to some functions of products of spectral radii of the mean matrices. This paper complements a former result [arXiv: 2007.07840] which requires in addition a condition $\forall k\ge1,\rm{det}(M_k)<-\varepsilon$ for some $\varep…
Local explosions and extinction in continuous-state branching processes with logistic competition
本文通过对偶方法研究具有逻辑竞争的连续状态分支过程的局部爆炸与灭绝，引入双对偶过程建立联系，并利用一维扩散理论转移结果。
We consider Galton-Watson branching processes with countable typeset $\mathcal{X}$. We study the vectors ${\bf q}(A)=(q_x(A))_{x\in\mathcal{X}}$ recording the conditional probabilities of extinction in subsets of types $A\subseteq \mathcal{X}$, given that the type of the initial individual is $x$. We first investigate the location of the vectors ${\bf q}(A)$ in the set of fixed points of the progeny generating vector and prove that $q_x(\{x\})$ is larger than or equal to the $x$th entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for $q_x(A)< q_x…
Stationary measures and the continuous-state branching process conditioned on extinction
本文研究了几乎必然灭绝的连续状态分支过程，通过Lévy过程尺度函数表示平稳测度，并证明了在临界与次临界情形下条件灭绝过程的极限定理及非退化极限分布。
We consider a spectrally positive Lévy process $X$ that does not drift to $+\infty$, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process \cite{GaJa98,Lam10}. We denote by $I$ the past infimum process defined for each $t\geq 0$ by $I_t:= \inf_{[0,t]} X$ and we let $γ$ be the unique time at which the excursion of the reflected process $X-I$ away from 0 attains its supremum. We prove that the pre-$γ$ and the post-$γ$ subpaths of this excursion are invariant under space-time reversal, which has several other consequ…
In this paper, we study the extinction time of logistic branching processes which are perturbed by an independent random environment driven by a Brownian motion. Our arguments use a Lamperti-type representation which is interesting on its own right and provides a one to one correspondence between the latter family of processes and the family of Feller diffusions which are perturbed by an independent spectrally positive Lévy process. When the independent random perturbation (of the Feller diffusion) is driven by a subordinator then the logistic branching processes in a Brownian environment conv…
We construct a continuous state branching process with immigration (CBI) whose immigration depends on the CBI itself and we recover a continuous state branching process (CB). This provides a dual construction of the pruning at nodes of CB introduced by the authors in a previous paper. This construction is a natural way to model neutral mutation. Using exponential formula, we compute the probability of extinction of the original type population in a critical or sub-critical quadratic branching, conditionally on the non extinction of the total population.
Strong survival and extinction for multitype branching processes via a new order for generating functions
本文提出多类型分支过程生成函数的一种新随机序，用以推导生存、强生存与灭绝判据，并推广 Moyal 关于生成函数不动点的结果。
Conditioning the logistic continuous-state branching process on non-extinction via its total progeny
本文研究了逻辑连续状态分支过程在非灭绝条件下的条件化，通过要求总种群数超过任意大的指数随机变量，构建了与Doob h变换相关的条件过程，并刻画了其有限寿命、随机方程表示及从零起始的性质。
A continuous-state polynomial branching process is constructed as the pathwise unique solution of a stochastic integral equation with absorbing boundary condition. The extinction and explosion probabilities and the mean extinction and explosion times are computed explicitly, which are also new in the classical branching case. We present necessary and sufficient conditions for the process to extinguish or explode in finite times. In the critical or subcritical case, we give a construction of the process coming down from infinity. Finally, it is shown that the continuous-state polynomial branchi…
We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schrödinger type operator. We first determine their growth rates on the survival event. We then establish the upper deviation for the maximal displacement under the possibility of extinction. Under the non-extinction condition, we further discuss the decay rate of the upper deviation probability and the population growth at the critical phase.
A branching process in a Markovian environment consists of an irreducible Markov chain on a set of "environments" together with an offspring distribution for each environment. At each time step the chain transitions to a new random environment, and one individual is replaced by a random number of offspring whose distribution depends on the new environment. We give a first moment condition that determines whether this process survives forever with positive probability. On the event of survival we prove a law of large numbers and a central limit theorem for the population size. We also define a …
A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process $Z_n$ with offspring distribution $Y$, there exists a random variable $X$ such that the probability $P(Z_n=0)$ of extinction of the $n$th generation in the branching process equals the value obtained by optimally stopping the sequence $X_1,...,X_n$, where these variables are i.i.d distributed as $X$. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple `stoppi…
This paper concerns the almost sure time dependent local extinction behavior for super-coalescing Brownian motion $X$ with $(1+β)$-stable branching and Lebesgue initial measure on $\bR$. We first give a representation of $X$ using excursions of a continuous state branching process and Arratia's coalescing Brownian flow. For any nonnegative, nondecreasing and right continuous function $g$, put τ:=\sup \{t\geq 0: X_t([-g(t),g(t)])>0 \}. We prove that $\bP\{τ=\infty\}=0$ or 1 according as the integral $\int_1^\infty g(t)t^{-1-1/β} dt$ is finite or infinite.
After generalizing the criteria introduced by Chen, we establish the necessary and sufficient conditions for the extinction, explosion and coming down from infinity of a continuous-state nonlinear Neveu's branching process.
In this paper we provide explicit representations of Laplace transforms of extinction time and total progeny of rough continuous-state branching processes introduced in [7]. Also, we show that their tail distributions are much fatter than those of Feller branching diffusions.
In this paper, we consider $n$-type Markov branching processes with immigration and resurrection. The uniqueness criteria are first established. Then, a new method is found and the explicit expression of extinction probability is successfully obtained in the absorption case, the mean extinction time is also given. The recurrence and ergodicity criteria are given if the state ${\bf 0}$ is not absorptive. Finally, if the resurrection rates are same as the immigration rates, the branching property and decay property are discussed in detail, it is shown that the process is a superimposition of a $…
The Erlang Weighted Tree, A New Branching Process
本文提出埃尔朗加权树（EWT）这一新型分支过程，研究其灭绝概率与增长率，证明灭绝概率为某算子的最小不动点，并给出灭绝概率等于1的充要条件为 Krein-Rutman 特征值 β0 ≤ 1。
Consider a catalytic super-Brownian motion $X=X^Γ$ with finite variance branching. Here `catalytic' means that branching of the reactant $X$ is only possible in the presence of some catalyst. Our intrinsic example of a catalyst is a stable random measure $Γ$ on $R$ of index $0< gamma <1$. Consequently, here the catalyst is located in a countable dense subset of $R$. Starting with a finite reactant mass $X_0$ supported by a compact set, $X$ is shown to die in finite time. Our probabilistic argument uses the idea of good and bad historical paths of reactant `particles' during time periods $[T_{n…
We study the asymptotic behavior of the probability of non extinction of a weakly subcritical multitype branching process in iid random environments. Under suitable assumptions, the survival probability is of order of $ρ^n n ^{-3/2}$ for some $ρ\in (0, 1)$ to specify.
Gelation in Vector Multiplicative Coalescence and Extinction in Multi-Type Poisson Branching Processes
本文建立了向量乘法凝聚过程与多类型泊松分支过程之间的等价关系，据此给出凝胶化的快速证明及灭绝概率的级数表达式，并利用随机图推导修正Smoluchowski方程的解。
A general model of catalytic branching process (CBP) with any finite number of catalysis centers in a discrete space is studied. More exactly, it is assumed that particles move in this space according to a specified Markov chain and they may produce offspring only in the presence of catalysts located at fixed points. The asymptotic (in time) behavior of the total number of particles as well as the local particles numbers is investigated. The problems of finding the global extinction probability and local extinction probability are solved. Necessary and sufficient conditions are established for…
We consider population-size-dependent branching processes (PSDBPs) which eventually become extinct with probability one. For these processes, we derive maximum likelihood estimators for the mean number of offspring born to individuals when the current population size is $z\geq 1$. As is standard in branching process theory, an asymptotic analysis of the estimators requires us to condition on non-extinction up to a finite generation $n$ and let $n\to\infty$; however, because the processes become extinct with probability one, we are able to demonstrate that our estimators do not satisfy the clas…
Under mild non-degeneracy assumptions on branching rates in each generation, we provide a criterion for almost-sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as time goes to infinity.
Tails of extinction time and maximal displacement for critical branching killed Lévy process
本文研究临界分支带杀Lévy过程的灭绝时间与最大位移尾渐近行为，推导了存活概率与尾概率的衰减速率，并得到相关标度极限的表示结果。
We study supercritical branching processes under the influence of an i.i.d. emigration component. We provide conditions, under which the lifetime of the process is finite respectively has a finite expectation. A new version of the Kesten-Stigum theorem is obtained and the extinction probability for a large initial population size is related to the tail behaviour of the emigration.
Conditions for almost sure extinction are studied in discrete time branching processes with an infinite number of types. It is not assumed that the expected number of children is a bounded function of the parent's type. There might also be no integer m such that there is a lower positive bound, uniform over the ancestor's type, for the probability that a population is extinct at the m-th generation. A weaker condition than the existence of such an m is seen to lead to extinction almost surely if the sequence of expected generation sizes does not tend to infinity. Some criteria for a positive p…
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