Processes In Random Environment Research Articles

Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment

Let $\xi=(\xi_t, t\ge 0)$ be a real-valued Levy process and define its associated exponential functional as follows \[ I_t(\xi):=\int_0^t \exp\{-\xi_s\}{\rm d} s, \qquad t\ge 0. \] Motivated by important applications to stochastic processes in random environment, we study the asymptotic behaviour of \[ \mathbb{E}\Big[F\big(I_t(\xi)\big)\Big] \qquad \textrm{as}\qquad t\to \infty, \] where $F$ is a non-increasing function with polynomial decay at infinity and under some exponential moment conditions on $\xi$. In particular, we find five different regimes that depend on the shape of the Laplace exponent of $\xi$. Our proof relies on a discretisation of the exponential functional $I_t(\xi)$ and is closely related to the behaviour of functionals of semi-direct products of random variables. We apply our main result to three {questions} associated to stochastic processes in random environment. We first consider the asymptotic behaviour of extinction and explosion for {stable} continuous state branching processes in a Levy random environment. Secondly, we {focus on} the asymptotic behaviour of the mean of a population model with competition in a Levy random environment and finally, we study the tail behaviour of the maximum of a diffusion process in a Levy random environment.

Branching within branching: A model for host–parasite co-evolution

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 branching-within-branching model and then focus on the property of parasite extinction. We establish equivalent conditions for almost sure extinction of parasites and find a strong relation of this event to the behavior of parasite multiplication along a randomly chosen cell line through the cell tree, which forms a branching process in random environment. We then focus on asymptotic results for relevant processes in the case when parasites survive. In particular, limit theorems for the processes of contaminated cells and of parasites are established by using martingale theory and the technique of size-biasing. The results for both processes are of Kesten–Stigum type by including equivalent integrability conditions for the martingale limits to be positive with positive probability. The case when these conditions fail is also studied. For the process of contaminated cells, we show that a proper Heyde–Seneta norming exists such that the limit is nondegenerate.

Random environment integer‐valued autoregressive process

An r states random environment integer‐valued autoregressive process of order 1, RrINAR(1), is introduced. Also, a random environment process is separately defined as a selection mechanism of differently parameterized geometric distributions, thus ensuring the non‐stationary nature of the RrNGINAR(1) model based on the negative binomial thinning. The distributional and correlation properties of this model are discussed, and the k‐step‐ahead conditional expectation and variance are derived. Yule–Walker estimators of model parameters are presented and their strong consistency is proved. The RrNGINAR(1) model motivation is justified on simulated samples and by its application to specific real‐life counting data.

Scaling limit of local time of Sinai’s random walk

We prove that the local times of a sequence of Sinai’s random walks converge to those of Brox’s diffusion by proper scaling. Our proof is based on the intrinsic branching structure of the random walk and the convergence of the branching processes in random environment.

A Study of Markov Branching Process with Varying Growth Rate

Branching processes in random environments occurs when the values of the reproduction probabilities and instantaneous reproduction rates are themselves sample paths of stochastic processes. These processes occur very naturally as models to describe the growth mechanism of biological systems. In this paper, we consider a Markov branching process with varying growth rates controlled by a random environment described by an alternating renewal process; we will derive expression for the mean number of the size of the population.

Limiting behaviors of bisexual branching processes in random environments

For a bisexual branching process in a random environment, the conditional mean growth rate per mating unit is introduced, and its related properties are studied, then the upper and lower bounds of conditional mean of the process are obtained. The limiting behaviors of the number of mating units in each generation normalized by these two bounds are respectively investigated. Particularly, the limiting behaviors of the number of females and males in each generation normalized by slightly modi ed bounds are discussed too, and equivalence theorems between the normalized number of mating units and that of females or males in each generation are obtained.

Quenched large deviations for multiscale diffusion processes in random environments

We consider multiple time scales systems of stochastic differential equations with small noise in random environments. We prove a quenched large deviations principle with explicit characterization of the action functional. The random medium is assumed to be stationary and ergodic. In the course of the proof we also prove related quenched ergodic theorems for controlled diffusion processes in random environments that are of independent interest. The proof relies entirely on probabilistic arguments, allowing to obtain detailed information on how the rare event occurs. We derive a control, equivalently a change of measure, that leads to the large deviations lower bound. This information on the change of measure can motivate the design of asymptotically efficient Monte Carlo importance sampling schemes for multiscale systems in random environments

On the scaling limits of Galton-Watson processes in varying environments

We establish a general sufficient condition for a sequence of Galton–Watson branching processes in varying environments to converge weakly. This condition extends previ- ous results by allowing offspring distributions to have infinite variance. Our assumptions are stated in terms of pointwise convergence of a triplet of two real- valued functions and a measure. The limiting process is characterized by a backwards integro-differential equation satisfied by its Laplace exponent, which generalizes the branching equation satisfied by continuous state branching processes. Several examples are discussed, namely branching processes in random environment, Feller diffusion in varying environments and branching processes with catastrophes.

Conditional log-Laplace functional for a class of branching processes in random environments

A conditional log-Laplace functional (CLLF) for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching and an interacting dynamic generated by the random environments. CLLF will play an important role in the investigation of branching processes and superprocesses with interaction.

Quantifying stochastic introgression processes in random environments with hazard rates

Introgression is the permanent incorporation of genes from the genome of one population into another. Previous studies have found that stochasticity in number of offspring, hybridisation, and environment are important aspects of introgression risk, but these factors have been studied separately. In this paper we extend the use of the hazard rate which we previously used to study effects of demographic stochasticity with repeated invasion attempts, to incorporate temporal environmental stochasticity. We find that introgression risk varies much in time, and in some periods it can be much enhanced in such environments. Furthermore, effects of plant life history parameters, such as flowering and survival probabilities, on hazard rates depend on characteristics of the environmental variation.

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.

Large deviations for the contact process in random environment

The asymptotic shape theorem for the contact process in random environment gives the existence of a norm $\mu$ on $\mathbb{R}^{d}$ such that the hitting time $t(x)$ is asymptotically equivalent to $\mu(x)$ when the contact process survives. We provide here exponential upper bounds for the probability of the event $\{\frac{t(x)}{\mu(x)}\notin[1-\varepsilon,1+\varepsilon]\}$; these bounds are optimal for independent random environment. As a special case, this gives the large deviation inequality for the contact process in a deterministic environment, which, as far as we know, has not been established yet.

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 ).

Jump-diffusion processes in random environments

In this paper we investigate jump-diffusion processes in random environments which are given as the weak solutions of SDEs. We formulate conditions ensuring existence and uniqueness in law of solutions. We investigate the Markov property. To prove uniqueness we solve a general martingale problem for càdlàg processes. This result is of independent interest. Application of our results to generalized exponential Lévy model are present in the last section.

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.

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.