Processes In Random Environment Research Articles

Properties of multitype subcritical branching processes in random environment

Abstract We study properties of a p-type subcritical branching process in random environment initiated at moment zero by a vector z = (z 1, .., z p ) of particles of different types. For p = 1 the class of processes we consider corresponds to the so-called strongly subcritical case. It is shown that the survival probability of this process up to moment n behaves as C(z)λ n for large n, where the parameters λ ∈ (0, 1) and C(z) ∈ (0, ∞) are explicitly described in terms of the characteristics of the process. We also demonstrate that the distribution of the number of particles of different types at moment n → ∞ (given its survival up to this moment) does not asymptotically depend on the number and types of particles initiated the process.

General Self-Similarity Properties for Markov Processes and Exponential Functionals of Lévy Processes

Positive self-similar Markov processes are positive Markov processes that satisfy the scaling property and it is known that they can be represented as the exponential of a time-changed Lévy process via Lamperti representation. In this work, we are interested in what happens if we consider Markov processes in dimension 1 or 2 that satisfy self-similarity properties of a more general form than a scaling property. We characterize them by proving a generalized Lamperti representation. Our results show that, in dimension 1, the classical Lamperti representation only needs to be slightly generalized. However, in dimension 2, our generalized Lamperti representation is much more different and involves the exponential functional of a bivariate Lévy process. We briefly discuss the complications that occur in higher dimensions. We present examples in dimensions 1, 2 and 3 that are built from growth-fragmentation, self-similar fragmentation and Continuous-state Branching processes in Random Environment. Some of our arguments apply in the context of a general state space and show that we can exhibit a topological group structure on the state space of a Markov process that satisfies general self-similarity properties, which allows to write a Lamperti-type representation for this process in terms of a Lévy process on the group.

Hydrodynamics for the partial exclusion process in random environment

In this paper, we introduce a random environment for the exclusion process in Zd obtained by assigning a maximal occupancy to each site. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we derive a quenched hydrodynamic limit in path space by strengthening the mild solution approach initiated in Nagy (2002) and Faggionato (2007). To this purpose, we prove, employing the technology developed for the random conductance model, a homogenization result in the form of an arbitrary starting point quenched invariance principle for a single particle in the same environment, which is a result of independent interest. The self-duality property of the partial exclusion process allows us to transfer this homogenization result to the particle system and, then, apply the tightness criterion in Redig et al. (2020).

Multitype weakly subcritical branching processes in random environment

Abstract A multi-type branching process evolving in a random environment generated by a sequence of independent identically distributed random variables is considered. The asymptotics of the survival probability of the process for a long time is found under the assumption that the matrices of the mean values of direct descendants have a common left eigenvector and the increment X of the associated random walk generated by the logarithms of the Perron roots of these matrices satisfies conditions E X < 0 and E XeX > 0.

Critical branching processes in random environment with immigration: survival of a single family

We consider a critical branching process in an i.i.d. random environment, in which one immigrant arrives at each generation. We are interested in the event \(\mathcal {A}_{i}(n)\) that all individuals alive at time n are offsprings of the immigrant which joined the population at time i. We study the asymptotic probability of this extreme event 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). In order to do so, we establish some limit theorems for random walks conditioned to stay positive or nonnegative, which are of independent interest.

Subcritical Branching Processes in Random Environment with Immigration: Survival of a Single Family

We consider a subcritical branching process in an independent and identically distributed (i.i.d.) random environment, where one immigrant arrives at each generation. We consider the event $\mathca...

Multitype branching processes in random environment

В работе дан обзор результатов по теории многотипных ветвящихся процессов, эволюционирующих в случайной среде. Библиография: 104 названия.

Geometric adaptive Monte Carlo in random environment

Manifold Markov chain Monte Carlo algorithms have been introduced to sample more effectively from challenging target densities exhibiting multiple modes or strong correlations. Such algorithms exploit the local geometry of the parameter space, thus enabling chains to achieve a faster convergence rate when measured in number of steps. However, acquiring local geometric information can often increase computational complexity per step to the extent that sampling from high-dimensional targets becomes inefficient in terms of total computational time. This paper analyzes the computational complexity of manifold Langevin Monte Carlo and proposes a geometric adaptive Monte Carlo sampler aimed at balancing the benefits of exploiting local geometry with computational cost to achieve a high effective sample size for a given computational cost. The suggested sampler is a discrete-time stochastic process in random environment. The random environment allows to switch between local geometric and adaptive proposal kernels with the help of a schedule. An exponential schedule is put forward that enables more frequent use of geometric information in early transient phases of the chain, while saving computational time in late stationary phases. The average complexity can be manually set depending on the need for geometric exploitation posed by the underlying model.

On generalized random environment INAR models of higher order: Estimation of random environment states

The behavior of a generalized random environment integer-valued autoregressive model of higher order with geometric marginal distribution and negative binomial thinning operator is dictated by a realization {zn}?,n=1 of an auxiliary Markov chain called random environment process. Element zn represents a state of the environment in moment n ? N and determines all parameters of the model in that moment. In order to apply the model, one first needs to estimate {zn}?,n=1, which was so far done by K-means data clustering. We argue that this approach ignores some information and performs poorly in certain situations. We propose a new method for estimating {zn}?,n=1, which includes the data transformation preceding the clustering, in order to reduce the information loss. To confirm its efficiency, we compare this new approach with the usual one when applied on the simulated and the real-life data, and notice all the benefits obtained from our method.

On the extinction of continuous-state branching processes in random environments

This paper establishes a model of continuous-state branching processes with time inhomogeneous competition in Levy random environments. Some results on extinction are presented, including the distribution of the extinction time, the limiting distribution conditioned on large extinction times and the asymptotic behavior near extinction. This paper also provides a new time-space transformation which can be used for further exploration in similar models. The results are applied to an epidemic model to describe the dynamics of infectious population and a virus model to describe the dynamics of viral load.

A Phase Transition for Large Values of Bifurcating Autoregressive Models

We describe the asymptotic behavior of the number $$Z_n[a_n,\infty )$$ of individuals with a large value in a stable bifurcating autoregressive process, where $$a_n\rightarrow \infty $$ . The study of the associated first moment is equivalent to the annealed large deviation problem of an autoregressive process in a random environment. The trajectorial behavior of $$Z_n[a_n,\infty )$$ is obtained by the study of the ancestral paths corresponding to the large deviation event together with the environment of the process. This study of large deviations of autoregressive processes in random environment is of independent interest and achieved first. The estimates for bifurcating autoregressive process involve then a law of large numbers for non-homogenous trees. Two regimes appear in the stable case, depending on whether one of the autoregressive parameters is greater than 1 or not. It yields different asymptotic behaviors for large local densities and maximal value of the bifurcating autoregressive process.

Random environment binomial thinning integer-valued autoregressive process with Poisson or geometric marginal

To predict time series of counts with small values and remarkable fluctuations, an available model is the $r$ states random environment process based on the negative binomial thinning operator and the geometric marginal. However, we argue that the aforementioned model may suffer from the following two drawbacks. First, under the condition of no prior information, the overdispersed property of the geometric distribution may cause the predictions fluctuate greatly. Second, because of the constraints on the model parameters, some estimated parameters are close to zero in real-data examples, which may not objectively reveal the correlation relationship. For the first drawback, an $r$ states random environment process based on the binomial thinning operator and the Poisson marginal is introduced. For the second drawback, we propose a generalized $r$ states random environment integer-valued autoregressive model based on the binomial thinning operator to model fluctuations of data. Yule–Walker and conditional maximum likelihood estimates are considered and their performances are assessed via simulation studies. Two real-data sets are conducted to illustrate the better performances of the proposed models compared with some existing models.

Branching Processes in Random Environment with Sibling Dependence

We consider a population of particles with unit lifetime. Dying, each particle produces offspring whose size depends on the random environment specifying the reproduction law of all particles of the given generation and on the number of relatives of the particle. We study the asymptotic behavior of the survival probability of the population up to a distant moment n under some restrictions on the properties of the environment and family ties and prove a limit theorem for the number of particles in such processes given the respective populations survive for a long time.

Subcritical Branching Processes in Random Environment with Immigration Stopped at Zero

We consider the 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 change of measure and some conditional limit theorems for random walks.

The Survival Probability for a Class of Multitype Subcritical Branching Processes in Random Environment

The asymptotic behavior of the survival probability for multi-type branching processes in a random environment is studied. In the case where all particles are of one type, the class of processes under consideration corresponds to intermediately subcritical processes. Under fairly general assumptions on the form of the generating functions of the laws of reproduction of particles, it is proved that the survival probability at a remote instant n of time for a process that started at the zero instant of time from one particle of any type is of the order of λnn−1/2, where λ ∈ (0, 1) is a constant defined in terms of the Lyapunov exponent for products of the mean-value matrices of the laws of reproduction of particles.

Recurrence of direct products of diffusion processes in random media having zero potentials

In this paper, we introduce an index which measures the strength of recurrence of symmetric Markov processes, and give some sufficient conditions for recurrence of direct products of symmetric diffusion processes. The index is given by the Dirichlet forms of the Markov processes. Moreover, as an application, we prove the recurrence of some multi-dimensional diffusion processes in random environments including zero potentials.

Critical branching processes in random environment and Cauchy domain of attraction

We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment. We assume that the random walk associated with the branching process is oscillating and satisfies a Spitzer condition $\mathbf{P}(S_{n}>0)\rightarrow \rho ,\ n\rightarrow \infty $, which is a standard condition in fluctuation theory of random walks. Unlike the previously studied case $\rho \in (0,1)$, we investigate the case where the offspring distribution is in the domain of attraction of a stable law with parameter $1$, which implies that $\rho =0$ or $1$. We find the asymptotic behaviour of the survival probability of the population in these two cases.

Representations of the Vertex Reinforced Jump Process as a mixture of Markov processes on $\mathbb {Z}^{d}$ and infinite trees

This paper concerns the Vertex Reinforced Jump Process (VRJP) and its representations as a Markov process in random environment. In [21], it was shown that the VRJP on finite graphs, under a certain time rescaling, has the distribution of a mixture of Markov jump processes. This representation was extended to infinite graphs in [23], by introducing a random potential $\beta $. In this paper, we show that all possible representations of the VRJP as a mixture of Markov processes can be expressed in a similar form as in [23], using the random field $\beta $ and harmonic functions for an associated operator $H_\beta $. This allows to show that the VRJP on $\mathbb {Z}^{d}$ (with certain initial conditions) has a unique representation, by proving that an associated Martin boundary is trivial. Moreover, on infinite trees, we construct a family of representations, that are all different when the VRJP is transient and the tree is $d$-regular (with $d\geq 3$).

Some properties of two classes of stochastic processes with asymptotic perturbation

In this paper, we consider two kinds of stochastic models with asymptotic perturbation. One is the non-nearest neighbor random walk with asymptotic perturbation, for which we consider the transience and recurrence of the random walk.We give the criterion for positive recurrent of the random walk. The other is the branching process in random environment with asymptotic perturbation,for which by analyzing the limit behavior of the associated random walk, we give the criterion of extinction and non-extinction of the branching process.

On the exit time and stochastic homogenization of isotropic diffusions in large domains

On effectue l’homogeneisation stochastique d’une certaine classe d’equations elliptiques et paraboliques. Ces equations decrivent la duree de vie, dans des domaines grands, de processus de diffusion stationnaire en environnement aleatoire qui sont des petites perturbations statistiquement isotropes du mouvement brownien, en dimension au moins trois. On demontre que l’homogeneisation a lieu a vitesse algebrique. De tels processus ont ete etudies dans un cadre continu en premier lieu par Snitzman et Zeitouni (Invent. Math. 164 (2006) 455–567), sur les resultats desquels le present travail s’appuie fortement.