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.     

随机环境中r维分支链的协方差计算

定义了各种条件多元概率母函数，并利用条件多元概率母函数这一强有力工具研究随机环境中r-维分支链的性质，并给出了其协方差阵的精确计算公式．     

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 EXISTENCE AND MOMENTS OF CANONICAL BRANCHING CHAIN IN RANDOM ENVIRONMENT

The concepts of branching chain in random environmnet and canonical branch-ing chain in random environment are introduced. Moreover the existence of these chains is proved. Finally the exact formulas of mathematical expectation and variance of branching chain in random environment are also given.     

THE CONSTRUCTION OF MULTITYPE CANONICAL MARKOV BRANCHING CHAINS IN RANDOM ENVIRONMENTS

The investigation for branching processes has a long history by their strong physics background, but only a few authors have investigated the branching processes in random environments. First of all, the author introduces the concepts of the multitype canonical Markov branching chain in random environment (CMBCRE) and multitype Markov branching chain in random environment (MBCRE) and proved that CMBCRE must be MBCRE, and any MBCRE must be equivalent to another CMBCRE in distribution. The main results of this article are the construction of CMBCRE and some of its probability properties.     

ON MARKOV CHAINS IN SPACE-TIME RANDOM ENVIRONMENTS

In Section 1, the authors establish the models of two kinds of Markov chains in space-time random environments (MCSTRE and MCSTRE(+)) with abstract state space. In Section 2, the authors construct a MCSTRE and a MCSTRE(+) by an initial distribution Φ and a random Markov kernel (RMK) p(γ). In Section 3, the authors es-tablish several equivalence theorems on MCSTRE and MCSTRE(+). Finally, the authors give two very important examples of MCMSTRE, the random walk in spce-time random environment and the Markov br...     

具有随机控制函数的受控分枝过程

本文研究了具有随机控制函数的受控分枝过程的各类概率母函数间的关系.利用对概率母函数求导的方法,获得了过程的期望和方差,推广了该过程概率母函数和矩量的一些基本结果.     

随机环境中分枝q-过程

引进了机环境中一致马氏过程,随机分枝q-矩阵和随机环境中分枝q-过程.给了随机环境中分枝q-过程存在性和唯一性的充分条件.最后证明了任意随机分枝转移密度矩阵都是零流入的.     

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.     

Branching Random Walk in a Catalytic Medium. I. Basic Equations

We consider a continuous-time branching random walk on the integer lattice ^d (d 1 ) with a finite number of branching sources, or catalysts. The random walk is assumed to be spatially homogeneous and irreducible. The branching mechanism at each catalyst, being independent of the random walk, is governed by a Markov branching process. The quantities of interest are the local numbers of particles (at each site) and the total population size. In the present paper, we derive and analyze the Kolmogorov type backward equations for the corresponding Laplace generating functions and also for the successive integer moments and the process extinction probability. In particular, existence and uniqueness theorems are proved and the problem of explosion is studied in some detail. We then rewrite these equations in the form of integral equations of renewal type, which may serve as a convenient tool for the study of the process long-time behavior. The paper also provides a technical foundation to some results published before without detailed proofs.     

Bisexual Galton-Watson Branching Processes in Random Environments

In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment proccss. Some results for the probability generating functions associated with the process are obtained and sufficient conditions for certain extinction and for non-certain extinction are established.     

随机环境中多物种分枝游动质点密度矩阵的极限分布

本文研究了随机环境中的多物种分枝游动于时刻k,位置x的质点密度矩阵序列{M~(k)(x)}k>1的极限分布。我们在证明了M~(k)(x),k>1,x∈Z是k个独立同分布的矩阵值随机元的乘积的基础上,主要证明了随机序列{logM_(ij)~(k)(x)}k>1依某种意义规范后是渐近正态的。     

Second order limit theorems for the Markov branching process in random environments

In a Markov branching process with random environments, limiting fluctuations of the population size arise from the changing environment, which causes random variation of the ‘deterministic’ population prediction, and from the stochastic wobble around this ‘deterministic’ mean, which is apparent in the ordinary Markov branching process. If the random environment is generated by a suitable stationary process, the first variation typically swamps the second kind. In this paper, environmental processes are considered which, in contrast, lead to sampling and environmental fluctuation of comparable magnitude. The method makes little use either of stationarity or of the branching property, and is amenable to some generalization away from the Markov branching process.     

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.     

Galton-Watson分支过程的谱半径的概率刻画

谱半径是不可约马尔可夫链的一个很重要的特征数字.Galton-Watson分支过程是一类特殊的马尔可夫链,我们已经证明了在不可约的条件下,Galton-Watson分支过程的谱半径等于其对应的概率母函数f(s)在灭绝概率q点的导数值.本文主要从理论上刻画从过程的任何状态逃离速度的Galton-Watson分支过程的谱半径的概率意义.     

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.     

Scenery reconstruction with branching random walk

We study the problem of scenery reconstruction in arbitrary dimension using observations registered in boxes of size k (for k fixed), seen along a branching random walk. We prove that, using a large enough k for almost all the realizations of the branching random walk, almost all sceneries can be reconstructed up to equivalence.     

随机环境下上临界分值过程的偏差不等式

令$\{Z_{n}, n\ge 0\}$为独立同分布随机环境下的上临界分支过程$\xi=(\xi_n)_{n\geq 0}$.本文给出了$\ln (Z_{n+n_{0}}/Z_{n_{0}})$的一些偏差不等式及其在构造置信区间上的一些应用.     

Harmonic moments of branching processes in random environments

We consider harmonic moments of branching processes in general random environments. For a sequence of square integrable random variables, we give some conditions such that there is a positive constant c that every variable in this sequence belong to Ac or A1c uniformly.