THE BRANCHING CHAIN WITH DRIFT IN SPACE-TIME RANDOM ENVIRONMENT （I）： MODEL,MARKOV PROPERTY,MOMENTS

There are three parts in this article. In Section 1, we establish the model of branching chain with drift in space-time random environment （BCDSTRE）, i.e., the coupling of branching chain and random walk. In Section 2, we prove that any BCDSTRE must be a Markov chain in time random environment when we consider the distribution of the particles in space as a random element. In Section 3, we calculate the first-order moments and the second-order moments of BCDSTRE.     

The limit theorems for random walk with state space ℝ in a space-time random environment

We consider a discrete time random environment. We state that when the random walk on real number space in a environment is i.i.d., under the law, the law of large numbers, iterated law and CLT of the process are correct space-time random marginal annealed Using a martingale approach, we also state an a.s. invariance principle for random walks in general random environment whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.     

Weighted moments for a supercritical branching process in a varying or random environment

Let W be the limit of the normalized population size of a supercritical branching process in a varying or random environment. By an elementary method, we find sufficient conditions under which W has finite weighted moments of the form EWpl(W), where p > 1, l 0 is a concave or slowly varying function.     

Infinitely dimensional control Markov branching chains in random environments

First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional generating functionals and random Markov transition functions of such chains and investigate their branching property. Base on these concepts we calculate the moments of the β-MBCRE and obtain the main results of this paper such as extinction probabilities, polarization and proliferation rate. Finally we discuss the classification of β-MBCRE according to the different standards.     

A random walk with a branching system in random environments

We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on Z with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.     

A class of two-sex branching processes with reproduction phase in a random environment

We model the demographic dynamics of populations with sexual reproduction where the reproduction phase occurs in a non-predictable environment and we assume the immigration/out-migration of mating units in the population. We introduce a general class of two-sex branching processes where, in each generation, the number of mating units which take part in the reproduction phase is randomly determined and the offspring probability distribution changes over time in a random environment. We provide several probabilistic results about the limit behaviour of populations whose dynamics is modelled by such a class of stochastic processes. In particular, we provide sufficient conditions for the almost sure extinction of the population or for its survival with a positive probability. As illustration, we include some simulated examples.     

随机环境中带移民的上临界分枝过程的Berry-Esseen界

考虑独立同分布的随机环境中带移民的上临界分枝过程(Zn).应用(Zn)与随机环境中不带移民分枝过程的联系,以及与相应随机游动的联系,在一些适当的矩条件下,本文证明关于log Zn的中心极限定理的Berry-Esseen界.     

The existence and uniqueness of<Emphasis Type="Italic">q</Emphasis>-process in random environment

We introduce some basic concepts such as random (sub-)transition function, q-function in random environment, g-process in random environment and some basic lemmas. For any continuous g-function in random environment, we prove that the g-process in random environment always exists, and that any g-process in random environment satisfies the random Kolmogorov backward equation and the minimal g-process in random environment always exists. When g is a continuous and conservative g-function in random environment, the necessary and sufficient conditions for the uniqueness of g-process in random environment are given. Finally the special cases, homogeneous random transition functions and homogeneous g-processes in random environments are considered.     

Central limit theorems for a branching random walk with a random environment in time

We consider a branching random walk with a random environment in time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The environment is supposed to be independent and identically distributed. For A ?, let Z_n(A) be the number of particles of generation n located in A. We show central limit theorems for the counting measure Z_n(·) with appropriate normalization.     

Age-dependent branching processes in random environments

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ0,ξ1,...) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξn) on R , and reproduce independently new particles according to a probability law p(ξn) on N. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean EξZ(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.     

随机环境中具有随机控制函数的受控分枝过程

对随机环境中具有随机控制函数的受控分枝过程进行了更为详尽的概率描述和直观解释;证明了此过程是时齐马氏链和随机环境中的马氏链,并对其概率母函数及矩量进行了讨论。     

The construction of markov processes in random environments and the equivalence theorems

In sec.1, we introduce several basic concepts such as random transition function, p-m process and Markov process in random environment and give some examples to construct a random transition function from a non-homogeneous density function. In sec. 2, we construct the Markov process in random enviromment and skew product Markov process by p -m process and investigate the properties of Markov process in random environment and the original process and environment process and skew product process. In sec. 3, we give several equivalence theorems on Markov process in random environment.     

THE CONSTRUCTION OF DENUMERABLE q-PROCESSES IN RANDOM ENVIRONMENTS SATISFYING （F） OR （B）

This article is a continuation of[9].Based on the discussion of random Kolmogorov forward（backward）equations,for any given q-matrix in random environment, Q（θ）=（q（θ;x,y）,x,y∈X）,an infinite class of q-processes in random environments satisfying the random Kolmogorov forward（backward）equation is constructed.Moreover, under some conditions,all the q-processes in random environments satisfying the random Kolmogorov forward（backward）equation are constructed.