Branching Process In Random Environment Research Articles

Recurrence and Transience of Critical Branching Processes in Random Environment with Immigration and an Application to Excited Random Walks

We establish recurrence and transience criteria for critical branching processes in random environments with immigration. These results are then applied to the recurrence and transience of a recurrent random walk in a random environment on ℤ disturbed by cookies inducing a drift to the right of strength 1.

Recurrence and Transience of Critical Branching Processes in Random Environment with Immigration and an Application to Excited Random Walks

We establish recurrence and transience criteria for critical branching processes in random environments with immigration. These results are then applied to the recurrence and transience of a recurrent random walk in a random environment on ℤ disturbed by cookies inducing a drift to the right of strength 1.

Small positive values for supercritical branching processes in random environment

Les processus de branchement en environnement aleatoire $(Z_{n}\colon n\geq0)$ sont une generalisation des processus de Galton Watson ou a chaque generation, la reproduction est choisie de maniere i.i.d. Dans le regime surcritique, ces processus survivent avec probabilite positive et croissent alors geometriquement. Ce papier considere l’evenement rare ou le processus prend des valeurs non nulles mais bornees en temps long. Nous decrivons ainsi le comportement asymptotique de $P(1\leq Z_{n}\leq k\vert Z_{0}=i)$ quand $n\rightarrow\infty$. Plus precisement, nous caracterisons la vitesse exponentielle alaquelle $\mathbb{P}(Z_{n}=k\vert Z_{0}=i)$ tend vers zero en utilisant une representation en epine due a Geiger. Nous donnons alors des bornes pour cette vitesse. Si la loi de reproduction est lineaire fractionnaire, la vitesse devient plus explicite et deux regimes apparaissent. Nous montrons par ailleurs que ces regimes affectent le comportement asymptotique de l’ancetre commun le plus recent de la population en vie a l’instant n quand cette derniere est conditionnee a prendre de petites valeurs en temps long.

Asymptotic properties of supercritical branching processes in random environments

We consider a supercritical branching process (Z n ) in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale W n = Z n / $\mathbb{E}$ [Z n |ξ], the convergence rates of W − W n (by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in L p , and the convergence in probability), and limit theorems (such as central limit theorems, moderate and large deviations principles) on (log Z n ).

Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions

In the present paper, we characterize the behavior of supercritical branching processes in random environment with linear fractional offspring distributions, conditioned on having small, but positive values at some large generation. As it has been noticed in previous works, there is a phase transition in the behavior of the process. Here, we examine the strongly and intermediately supercritical regimes The main result is a conditional limit theorem for the rescaled associated random walk in the intermediately case.

Conditional limit theorems for intermediately subcritical branching processes in random environment

Nous considérons un processus de branchement dans un environnement aléatoire dont la distribution des enfants des individus varie aléatoirement de façon indépendante d’une génération à l’autre. Dans le régime sous critique, une transition de phase apparaît. Cet article est consacré à l’étude de la région proche de la transition. Nous étudions le comportement asymptotique de la probabilité de survie ainsi que la taille de la population et la forme de l’environnement aléatoire sous la condition de non-extinction. Nous montrons finalement que conditionnée à la non-extinction, la population alterne des périodes de petite et de grande taille. Ce type de comportement apparaît sous la mesure moyennée uniquement dans ce régime sous critique proche de la transition.

Functional limit theorems for high-level subcritical branching processes in random environment

AbstractThe paper is concerned with subcritical branching process in random environment. It is assumed that the moment-generating function of steps of the associated random walk is equal to 1 for some positive value of the argument. Functional limit theorems for sizes of various generations and passage times to various levels are put forward.

Simulations and a conditional limit theorem for intermediately subcritical branching processes in random environment

Intermediately subcritical branching processes in random environment are at the borderline between two subcritical regimes and exhibit 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.

Evolution of branching processes in a random environment

This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in independent and identically distributed random environments. This is a natural generalization of the time-inhomogeneous branching processes. The key assumptions of the family of population models in question are nonoverlapping generations and discrete time. The reader should be aware of the fact that there are many very interesting papers covering other issues in the theory of branching processes in random environments which are not mentioned here.

Lower large deviations for supercritical branching processes in random environment

Branching processes in random environment (Z n : n ≥ 0) are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical regime, the process survives with a positive probability and grows exponentially on the non-extinction event. We focus on rare events when the process takes positive values but lower than expected. More precisely, we are interested in the lower large deviations of Z, which means the asymptotic behavior of the probability {1 ≤ Z n ≤ exp(nθ)} as n → ∞. We provide an expression for the rate of decrease of this probability under some moment assumptions, which yields the rate function. With this result we generalize the lower large deviation theorem of Bansaye and Berestycki (2009) by considering processes where ℙ(Z 1 = 0 | Z 0 = 1) > 0 and also much weaker moment assumptions.

Perturbing transient random walk in a random environment with cookies of maximal strength

Nous considérons une marche aléatoire unidimensionnelle en environnement aléatoire qui est transiente à gauche. Cette marche est modifiée par des cookies qui induisent une dérive vers la droite. Le nombre de cookies par site est i.i.d. et indépendant de l’environnement. Des critères pour la récurrence et la transience de la marche sont obtenus. Pour cela, nous utilisons des processus de branchement sous-critiques en environnement aléatoire avec immigration et nous formulons des critères de récurrence et de transience pour ces processus.

Lower deviations of branching processes in random environment with geometrical offspring distributions

We consider the branching processes in random environment. In this paper, we deal with the case of environments which are chosen stationary and ergodic from the finite set of geometrical offspring distributions. We denote by Zn the population at the n-th generation. We show that the large deviation principle holds with a certain rate function for the total population when the environment satisfies some conditions. Also, we will show that the trajectory t→logZntn,t∈[0,1] converges to a deterministic function uniformly in probability conditioned on {0<Zn≤ecn}.

On the extinction of continuous state branching processes with catastrophes

We consider continuous state branching processes (CSBP's) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a Levy process with bounded variation paths. We construct the associated class of processes 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 make appear new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish when it becomes extinct. For a class of processes for which extinction and absorption coincide (including the $\alpha$ stable CSBP's plus a drift), we determine the speed of extinction. Four regimes appear, as in the case of branching processes in random environment in discrete time and space.The proofs rely on a fine study of the asymptotic behavior of exponential functionals of Levy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection.

Subcritical branching processes in a random environment without the Cramer condition

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 beginning of the process.

Polling Systems and Multitype Branching Processes in Random Environment with Final Product

Using methods of multitype branching processes in random environment counted by random characteristics, we study the tail distribution of busy periods and some other characteristics of the branching-type polling systems in which the service disciplines, input parameters, and service time distributions are changing in a random manner.

Phenotypic diversity and population growth in a fluctuating environment

Organisms adapt to fluctuating environments by regulating their dynamics, and by adjusting their phenotypes to environmental changes. We model population growth using multitype branching processes in random environments, where the offspring distribution of some organism having trait t ∈ in environment e ∈ ε is given by some (fixed) distribution ϒt,e on ℕ. Then, the phenotypes are attributed using a distribution (strategy) πt,e on the trait space . We look for the optimal strategy πt,e, t ∈ , e ∈ ε, maximizing the net growth rate or Lyapounov exponent, and characterize the set of optimal strategies. This is considered for various models of interest in biology: hereditary versus nonhereditary strategies and strategies involving or not involving a sensing mechanism. Our main results are obtained in the setting of nonhereditary strategies: thanks to a reduction to simple branching processes in a random environment, we derive an exact expression for the net growth rate and a characterization of optimal strategies. We also focus on typical genealogies, that is, we consider the problem of finding the typical lineage of a randomly chosen organism.

Phenotypic diversity and population growth in a fluctuating environment

Organisms adapt to fluctuating environments by regulating their dynamics, and by adjusting their phenotypes to environmental changes. We model population growth using multitype branching processes in random environments, where the offspring distribution of some organism having trait t ∈ in environment e ∈ ε is given by some (fixed) distribution ϒ t,e on ℕ. Then, the phenotypes are attributed using a distribution (strategy) π t,e on the trait space . We look for the optimal strategy π t,e , t ∈ , e ∈ ε, maximizing the net growth rate or Lyapounov exponent, and characterize the set of optimal strategies. This is considered for various models of interest in biology: hereditary versus nonhereditary strategies and strategies involving or not involving a sensing mechanism. Our main results are obtained in the setting of nonhereditary strategies: thanks to a reduction to simple branching processes in a random environment, we derive an exact expression for the net growth rate and a characterization of optimal strategies. We also focus on typical genealogies, that is, we consider the problem of finding the typical lineage of a randomly chosen organism.

Upper large deviations for Branching Processes in Random Environment with heavy tails

Branching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where 'in each generation' the reproduction law is picked randomly in an i.i.d. manner. The associated random walk of the environment has increments distributed like the logarithmic mean of the offspring distributions. This random walk plays a key role in the asymptotic behavior. In this paper, we study the upper large deviations of the BPRE $Z$ when the reproduction law may have heavy tails. More precisely, we obtain an expression for the limit of $-\log \mathbb{P}(Z_n\geq \exp(\theta n))/n$ when $n\rightarrow \infty$. It depends on the rate function of the associated random walk of the environment, the logarithmic cost of survival $\gamma:=-\lim_{n\rightarrow\infty} \log \mathbb{P}(Z_n \gt 0)/n$ and the polynomial rate of decay $\beta$ of the tail distribution of $Z_1$. This rate function can be interpreted as the optimal way to reach a given "large" value. We then compute the rate function when the reproduction law does not have heavy tails. Our results generalize the results of Böinghoff & Kersting (2009) and Bansaye & Berestycki (2008) for upper large deviations. Finally, we derive the upper large deviations for the Galton-Watson processes with heavy tails.

Supercritical branching diffusions in random environment

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 phase transition of subcritical branching processes in random environment.

Limit Theorems for Weakly Subcritical Branching Processes in Random Environment

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 nonextinction. Also a functional limit theorem is proved, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.