The asymptotic behavior of a subcritical Branching Process in Random Environment (BPRE) starting with several particles depends on whether the BPRE is strongly subcritical (SS), intermediate subcritical (IS) or weakly subcritical (WS). %Descendances of particles for BPRE are not independent. In the (SS+IS) case, the asymptotic probability of survival is proportional to the initial number of particles, and conditionally on the survival of the population, only one initial particle survives $a.s.$ These two properties do not hold in the (WS) case and different asymptotics are established, which require new results on random walks with negative drift. We provide an interpretation of these results by characterizing the sequence of environments selected when we condition on the survival of particles. This also raises the problem of the dependence of the Yaglom quasistationary distributions on the initial number of particles and the asymptotic behavior of the Q-process associated with a subcritical BPRE.
We consider a branching stable process with positive jumps, i.e. a continuous-time branching process in which the particles evolve independently as stable L{é}vy processes with positive jumps. Assuming the branching mechanism is critical or subcritical, we compute the asymptotics of the maximum location ever reached by a particle of the process.
Intermediately subcritical branching processes in random environment are at the borderline between two subcritical regimes and exhibit a particularly rich behavior. In this paper, we prove a functional limit theorem for these processes. It is discussed together with two other recently proved limit theorems for the intermediately subcritical case and illustrated by several computer simulations.
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.
We consider branching process evolving in i.i.d. random environment. It is assumed that the process is intermediately subcritical. We investigate the initial stage of the evolution of the process given its survival for a long time.
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.
We consider a subcritical Galton--Watson tree conditioned on having $n$ vertices with outdegree in a fixed set $Ω$. Under mild regularity assumptions we prove various limits related to the maximal offspring of a vertex as $n$ tends to infinity.
We consider the subcritical contact branching random walk on Zd in continuous time with the arbitrary number of offspring and with immigration. We prove the existence of the steady state (statistical equilibrium).
We study the asymptotic behaviour of the survival probability of a multitype branching process in random environment. The class of processes we consider here corresponds, in the one-dimensional situation, to the strongly subcritical case. We also prove a conditional limit theorem describing the distribution of the number of particles in the process given its survival for a long time.
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 consider a slightly subcritical branching Brownian motion with absorption, where particles move as Brownian motions with drift $-\sqrt{2+2\varepsilon}$, undergo dyadic fission at rate $1$, and are killed when they reach the origin. We obtain a Yaglom type asymptotic result, showing that the long run expected number of particles conditioned on survival grows exponentially as $1/\sqrt{\varepsilon}$ as the process approaches criticality.
A subcritical branching process in random environment (BPRE) is considered whose associated random walk does not satisfy the Cramer condition. The asymptotics for the survival probability of the process is investigated, and a Yaglom type conditional limit theorem is proved for the number of particles up to moment $n$ given survival to this moment. Contrary to other types of subcritical BPRE, the limiting distribution is not discrete. We also show that the process survives for a long time owing to a single big jump of the associate random walk accompanied by a population explosion at the beginn…
We consider a critical branching process $Y_{n}$ in an i.i.d. random environment, in which one immigrant arrives at each generation. Let $% \mathcal{A}_{i}(n)$ be the event that all individuals alive at time $n$ are offspring of the immigrant which joined the population at time $i$. We study the conditional distribution of $Y_{n}$ given $\mathcal{A}_{i}(n)$ when $n$ is large and $i$ follows different asymptotics which may be related to $n$ ($% i$ fixed, close to $n$, or going to infinity but far from $n$).
Let $(Z_n)_{n\geqslant 0}$ be a branching process in a random environment defined by a Markov chain $(X_n)_{n\geqslant 0}$ with values in a finite state space $\mathbb X$ starting at $X_0=i \in\mathbb X$. We extend from the i.i.d. environment to the Markovian one the classical classification of the branching processes into critical and strongly, intermediate and weakly subcritical states. In all these cases, we study the asymptotic behaviour of the probability that $Z_n>0$ as $n\to+\infty$.
We consider subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods of such processes defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty again. We prove that the tail distribution decays with exponential rate. The main tools are the change of measure and some conditional limit theorems for random walks.
For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. For the subcritical regime a kind of phase transition appears. In this paper we study the intermediately subcritical case, which constitutes the borderline within this phase transition. We study the asymptotic behavior of the survival probability. Next the size of the population and the shape of the random environment conditioned on non-extinction is examined. Finally we show that conditioned on non-extinction…
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…
Subcritical catalytic branching random walk on d-dimensional lattice is studied. New theorems concerning the asymptotic behavior of distributions of local particles numbers are established. To prove the results different approaches are used including the connection between fractional moments of random variables and fractional derivatives of their Laplace transforms. In the previous papers on this subject only supercritical and critical regimes were investigated assuming finiteness of the first moment of offspring number and finiteness of the variance of offspring number, respectively. In the p…
We consider a branching system consisting of particles moving according to a Markov family in $\Rd$ and undergoing subcritical branching with a constant rate $V>0$. New particles immigrate to the system according to homogeneous space-time Poisson random field. The process of the fluctuations of the rescaled occupation time is studied with very mild assumptions on the Markov family. In this general setting a functional central limit theorem is proved. The subcriticality of the branching law is crucial for the limit behaviour and in a sense overwhelms the properties of the particles' motion. It …
Let $\{Z_n\}_{n\geq 0 }$ be a critical or subcritical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on $\mathbb{R}^d$. Denote by $R_n:=\sup\{u>0:Z_n(\{x\in\mathbb{R}^d:|x|<u\})=0\}$ the radius of the largest empty ball centered at the origin of $Z_n$. In this work, we prove that after suitable renormalization, $R_n$ converges in law to some non-degenerate distribution as $n\to\infty$. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random wal…
In this paper we consider two related stochastic models. The first one is a branching system consisting of particles moving according to a Markov family in R^d and undergoing subcritical branching with a constant rate of V>0. New particles immigrate to the system according to a homogeneous space time Poisson random field. The second model is the superprocess corresponding to the branching particle system. We study rescaled occupation time process and the process of its fluctuations with very mild assumptions on the Markov family. In the general setting a functional central limit theorem as wel…
For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, 'supercritical'. This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on non-extinction. Also a functional limit theorem is proven, which makes the conditional supercriticality manife…
We study an iterated temporal and contemporaneous aggregation of $N$ independent copies of a strongly stationary subcritical Galton-Watson branching process with regularly varying immigration having index $α\in (0, 2)$. Limits of finite dimensional distributions of appropriately centered and scaled aggregated partial sum processes are shown to exist when first taking the limit as $N \to \infty$ and then the time scale $n \to \infty$. The limit process is an $α$-stable process if $α\in (0, 1) \cup (1, 2)$, and a deterministic line with slope $1$ if $α= 1$.
We study the asymptotic behaviour of the survival probability of a multi-type branching processes in random environment. The class of processes we consider corresponds, in the one-dimensional situation, to the intermediately subcritical case. We show under rather general assumptions on the form of the offspring generating functions of particles that the probability of survival up to generation $n$ of the process initiated at moment zero by a single particle of any type is of order $λ^{n}n^{-1/2}$ for large $n,$ where $λ\in (0,1)$ is a constant specified by the Lyapunov exponent of the mean mat…
We study properties of a $p-$type subcritical branching process in random environment initiated at moment zero by a vector $\mathbf{z}=\left( z_{1},..,z_{p}\right) $\ of particles of different types. Assuming that the process belongs to the class of the so-called strongly subcritical processes we show that its survival probability to moment $n$\ behaves for large $n$\ as $C(\mathbf{z})λ^{n}$\ where $λ$\ is the upper Lyapunov exponent for the product of mean matrices of the process and $C(\mathbf{z})$% \ is an explicitly given constant. We also demonstrate that the limiting conditional distribu…
We study the density of the support of a dyadic $d$-dimensional branching Brownian motion (BBM) in subcritical balls in $\mathbb{R}^d$. Using elementary geometric arguments and an extension of a previous result on the probability of absence of the support of BBM in linearly moving balls of fixed size, we obtain sharp asymptotic results on the degree of density of the support of BBM in subcritical balls. As corollaries, we obtain almost sure results about the large-time behavior of $r(t)$-enlargement of the support of BBM when the shrinking radius $r(t)$ is decaying sufficiently slowly. As a by…
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 …
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…
We consider a subcritical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We consider the event $% \mathcal{A}_{i}(n)$ that all individuals alive at time $n$ are offspring of the immigrant which joined the population at time $i$ and investigate the asymptotic probability of this extreme event when $n\to\infty$ and $i$ is either fixed, or the difference $n-i$ is fixed, or $\min(i,n-i)\to\infty.$ To deduce the desired asymptotics we establish some limit theorems for random walks conditioned to be nonnegative or negative.
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 study the distribution of the maximal displacement of particles positions for the whole time of the population existence in the model of critical and subcritical catalytic branching random walk on Z. In particular, we prove that in the case of simple symmetric random walk on Z, the distribution of the maximal displacement has "a heavy tail" decreasing as a function of the power 1/2 or 1, when the branching process is critical or subcritical, respectively. These statements describe new effects which do not arise in the corresponding investigations of the maximal displacement of critical and …
Consider the population model with infinite size associated to subcritical continuous-state branching processes (CSBP). Individuals reproduce independently according to the same subcritical offspring distribution. We study the long-term behaviour of the ancestral lineages as time goes to the past and show that the flow of ancestral lineages, properly renormalized, converges almost surely to the inverse of a drift-free subordinator whose Laplace exponent is explicit in terms of the branching mechanism. We provide an interpretation in terms of the genealogy of the population. In particular, we s…
In this paper, we introduce a family of processes with values on the nonnegative integers that describes the dynamics of populations where individuals are allowed to have different types of interactions. The types of interactions that we consider include pairwise interaction, such as competition, annihilation and cooperation; and interaction among several individuals that can be considered as catastrophes. We call such families of processes \textit{branching processes with interactions}. Our aim is to study their long term behaviour under a specific regime of the pairwise interaction parameter…
Let $Z_{n}$ be the number of individuals in a subcritical BPRE evolving in the environment generated by iid probability distributions. Let $X$ be the logarithm of the expected offspring size per individual given the environment. Assuming that the density of $X$ has the form $$p_{X}(x)=x^{-β-1}l_{0}(x)e^{-ρx}$$ for some $β>2,$ a slowly varying function $l_{0}(x)$ and $ρ\in \left( 0,1\right),$ we find the asymptotic survival probability and prove a Yaglom type conditional limit theorem for the process. The survival probability decreases exponentially with an additional polynomial term related to…
We consider the diffusion approximation of branching processes in random environment (BPREs). This diffusion approximation is similar to and mathematically more tractable than BPREs. We obtain the exact asymptotic behavior of the survival probability. As in the case of BPREs, there is a phase transition in the subcritical regime due to different survival opportunities. In addition, we characterize the process conditioned to never go extinct and establish a backbone construction. In the strongly subcritical regime, mean offspring numbers are increased but still subcritical in the process condit…
We study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin. For each $n\in\mathbb{N},$ let $M_{n}$ be the rightmost position reached by the branching random walk up to generation $n$. Under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function, we show that there exists $ρ>1$ such that the function \[ g(c,n):=ρ^{cn} P(M_{n}\geq cn), \quad \mbox{for each }c>0 \mbox{ and } n\in\mathbb{N}, \] satisfies the following prop…
Consider a branching random walk on the real line with a killing barrier at zero: starting from a nonnegative point, particles reproduce and move independently, but are killed when they touch the negative half-line. The population of the killed branching random walk dies out almost surely in both critical and subcritical cases, where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed, and by critical case, when this speed is zero. We investigate the total progeny of the killed branching random walk and give their precise ta…
In this work we study the long-time behavior for subcritical measure-valued branching processes with immigration on the space of tempered measures. Under some reasonable assumptions on the spatial motion, the branching and immigration mechanisms, we prove the existence and uniqueness of an invariant measure for the corresponding Markov transition semigroup. Moreover, we show that it converges with exponential rate to the unique invariant measure in the Wasserstein distance as well as in a distance defined in terms of Laplace transforms. Finally, we consider an application of our results to sup…
The long-term behaviors of flows of continuous-state branching processes are characterized through subordinators and extremal processes. The extremal processes arise in the case of supercritical processes with infinite mean and of subcritical processes with infinite variation. The jumps of these extremal processes are interpreted as specific initial individuals whose progenies overwhelm the population. These individuals, which correspond to the records of a certain Poisson point process embedded in the flow, are called super-individuals. They radically increase the growth rate to $+\infty$ in …
We first study a model, introduced recently in \cite{ES}, of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no `obstacle' placed there. The obstacles appear at each site with probability $p\in [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical. Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $\mathbb P_p$-measure, as $n\to\infty$, (i) Critical case: P^ω(S…
We consider multitype branching processes arising in the study of random laminations of the disk. We classify these processes according to their subcritical or supercritical behavior and provide Kolmogorov-type estimates in the critical case corresponding to the random recursive lamination process of [1]. The proofs use the infinite dimensional Perron-Frobenius theory and quasi-stationary distributions.
