A note on multitype branching process with bounded immigration in random environment

In this paper, we study the total number of progeny, W, before regenerating of multitype branching process with immigration in random environment. We show that the tail probability of |W| is of order t-κ as t→∞, with κ some constant. As an application, we prove a stable law for (L-1) random walk in random environment, generalizing the stable law for the nearest random walk in random environment (see "Kesten, Kozlov, Spitzer: A limit law for random walk in a random environment. Compositio Math., 30, 145-168 (1975)").     

随机环境中迁入分枝过程的极限性质

在方差和均值有限的条件下,得到了随机环境中迁入分枝过程对应的规范化过程的几乎处处收敛性和L^2收敛性．这对于刻画过程本身的发散速度,具有重要的意义．     

On the trajectory of an individual chosen according to supercritical Gibbs measure in the branching random walk

Consider a branching random walk on the real line. Madaule (2016) showed the renormalized trajectory of an individual selected according to the critical Gibbs measure converges in law to a Brownian meander. Besides, Chen (2015) proved that the renormalized trajectory leading to the leftmost individual at time $n$ converges in law to a standard Brownian excursion. In this article, we prove that the renormalized trajectory of an individual selected according to a supercritical Gibbs measure also converges in law toward the Brownian excursion. Moreover, refinements of this results enables to express the probability for the trajectories of two individuals selected according to the Gibbs measure to have split before time $t$, partially answering a question of Derrida and Spohn (1988).     

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.     

The multitype branching random walk, II

Limit theorems for the multitype branching random walk as n → ∞ are given (n is the generation number) in the case in which the branching process has a mean matrix which is not positive regular. In particular, the existence of steady state distributions is proven in the subcritical case with immigration, and in the critical case with initial Poisson random fields of particles. In the supercritical case, analogues of the limit theorems of Kesten and Stigum are given.     

A random environment for linearly edge-reinforced random walks on infinite graphs

We consider linearly edge-reinforced random walk on an arbitrary locally finite connected graph. It is shown that the process has the same distribution as a mixture of reversible Markov chains, determined by time-independent strictly positive weights on the edges. Furthermore, we prove bounds for the random weights, uniform, among others, in the size of the graph.      

Upper limits of Sinai’s walk in random scenery

We consider Sinai’s walk in i.i.d. random scenery and focus our attention on a conjecture of Révész concerning the upper limits of Sinai’s walk in random scenery when the scenery is bounded from above. A close study of the competition between the concentration property for Sinai’s walk and negative values for the scenery enables us to prove that the conjecture is true if the scenery has “thin” negative tails and is false otherwise.     

一类随机环境下随机游动的常返性

就一类平稳环境θ下随机流动{Xn}n∈z 建立相应的Markov-双链{ηn}n∈z ={(xn,Tnθ)}n∈z ,并给出在该平稳环境θ下{xn}n∈z 为常返链的条件.     

Large deviations for hitting times of a random walk in random environment on a strip

We consider a random walk in random environment on a strip, which is transient to the right. The random environment is stationary and ergodic. By the constructed enlarged random environment which was first introduced by Goldsheid （2008）, we obtain the large deviations conditioned on the environment （in the quenched case） for the hitting times of the random walk.     

On the speed of a cookie random walk

We consider the model of the one-dimensional cookie random walk when the initial cookie distribution is spatially uniform and the number of cookies per site is finite. We give a criterion to decide whether the limiting speed of the walk is non-zero. In particular, we show that a positive speed may be obtained for just three cookies per site. We also prove a result on the continuity of the speed with respect to the initial cookie distribution.      

Velocity and Lyapounov exponents of some random walks in random environment

We express the asymptotic velocity of random walks in random environment satisfying Kalikow's condition in terms of the Lyapounov exponents which have previously been used in the context of large deviations.     

带形上随机环境中随机游动的内蕴分枝结构

揭示了带形上随机环境中随机游动的内蕴分枝结构一带移民的多物种分枝过程.利用内蕴分枝结构,可精确表达游动的首次击中时.给出了内蕴分枝结构的如下两个应用:(1)计算出首次击中时的均值,给出游动大数定律速度的显示表达,(2)得到从粒子角度看环境的马氏链不变测度的密度函数的显示表达,进而可用另一种\"站在粒子看环境\"的方法直接证明游动的大数定律.