Some Probability Properties of Random Walk in Time-Random Environment

A general formulation of the stochastic model for random walk in time-random environment and an equivalent definition is established in this paper. Moreover, some basic probability relations similar to the classical case which are very useful in the corresponding research of fractal properties are given. At the end, a typical example is provided to show the recurrence and transience. Key words random environment - random walk in timerandom environment - skew product Markov chain CLC number O 211.6 Foudation item: Supported by the National Natural Science Foundation of China (10371092) and Foundation of Wuhan University.Biography: Zhang Xiao-min (1977-), male, Ph. D candidate, research direction: stochastic processes and random fractal.

THE DIMENSIONS OF THE RANGE OF RANDOM WALKS IN TIME-RANDOM ENVIRONMENTS

Suppose {Xn} is a random walk in time-random environment with state space Z^d, ｜Xn｜ approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the range of {Xn} is any stability index α. Moreover, if the environment is stationary, a similar result for the lower bound of the discrete Hausdorff dimension is derived. Thus, the range is a fractal set for almost every environment.

ASYMPTOTIC PROPERTIES OF A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

We consider a branching random walk in an independent and identically distributed random environment ξ=(ξn) indexed by the time. Let W be the limit of the martingale Wn=∫e^-txZn(dx)/Eξ∫e^-txZn(dx), with Zn denoting the counting measure of particles of generation n, and Eξ the conditional expectation given the environment ξ. We find necessary and sufficient conditions for the existence of quenched moments and weighted moments of W, when W is non-degenerate.

CENTRAL LIMIT THEOREMS FOR A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

We consider a branching random walk with a random environment m 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 envi- ronment is supposed to be independent and identically distributed. For A C R, let Zn（A） be the number of particles of generation n located in A. We show central limit theorems for the counting measure Zn （-） with appropriate normalization.

RENEWAL THEOREM FOR(L,1)-RANDOM WALK IN RANDOM ENVIRONMENT

We consider a random walk on Z in random environment with possible jumps {-L,…, -1, 1}, in the case that the environment {ωi ： i ∈ Z} are i.i.d.. We establish the renewal theorem for the Markov chain of ＂the environment viewed from the particle＂ in both annealed probability and quenched probability, which generalize partially the results of Kesten （1977） and Lalley （1986） for the nearest random walk in random environment on Z, respectively. Our method is based on （L, 1）-RWRE formulated in Hong and Wang the intrinsic branching structure within the （2013）.

LYAPOUNOLYAPOUNOV EXPONENTS AND LAW OF LARGE NUMBERS FOR RANDOM WALK IN RANDOM ENVIRONMENT WITH HOLDING TIMES

In this article, the authors mainly discuss the law of large number under Kalikow＇s condition for multi-dimensional random walks in random environment with holding times. The authors give an expression to the escape speed of random walks in terms of the Lyapounov exponents, which have been precisely used in the context of large deviation.

ASYMPTOTIC BEHAVIOR FOR RANDOM WALK IN RANDOM ENVIRONMENT WITH HOLDING TIMES

In this article, we mainly discuss the asymptotic behavior for multi-dimensional continuous-time random walk in random environment with holding times. By constructing a renewal structure and using the point ＂environment viewed from the particle＂, under General Kalikow＇s Condition, we show the law of large numbers （LLN） and central limit theorem （CLT） for the escape speed of random walk.

Environment-dependent continuous time random walk

A generalized continuous time random walk model which is dependent on environmental damping is proposed in which the two key parameters of the usual random walk theory： the jumping distance and the waiting time, are replaced by two new ones： the pulse velocity and the flight time. The anomalous diffusion of a free particle which is characterized by the asymptotical mean square displacement （x^2（t）） - t^a is realized numerically and analysed theoretically, where the value of the power index a is in a region of 0 〈 a 〈 2. Particularly, the damping leads to a sub-diffusion when the impact velocities are drawn from a Gaussian density function and the super-diffusive effect is related to statistical extremes, which are called rare-though-dominant events.

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...

Exact Convergence Rate of the Local Limit Theorem for a Branching Random Walk in Z^(d)with a Random Environment in Time

Consider a branching random walk with a random environment in time in the d-dimensional integer lattice.The branching mechanism is governed by a supercritical branching process,and the particles perform a lazy random walk with an independent,non-identical increment distribution.For A■Z^(d),let Z_(n)(A)be the number of offsprings of generation n located in A.The exact convergence rate of the local limit theorem for the counting measure Z_(n)(·)is obtained.This partially extends the previous results for a simple branching random walk derived by Gao(2017,Stoch.Process Appl.).

Asymptotic Behavior for Random Walks in Time-Random Environment on Z^1

In this paper, we give a general model of random walks in time-random environment in any countable space. Moreover, when the environment is independently identically distributed, a recurrence-transience criterion and the law of large numbers are derived in the nearest-neighbor case on Z^1. At last, under regularity conditions, we prove that the RWIRE {Xn} on Z^1 satisfies a central limit theorem, which is similar to the corresponding results in the case of classical random walks.

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 ? 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.

Branching random walks with random environments in time

We consider a branching random walk on N with a random environment in time （denoted by ξ）. Let Zn be the counting measure of particles of generation n, and let Zn（t） be its Laplace transform. We show the convergence of the free energy n-llog Zn（t）, large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn（t）/E[Zn（t）ξ].

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.

Quenched moderate deviations principle for random walk in random environment

We derive a quenched moderate deviations principle for the one-dimensional nearest random walk in random environment,where the environment is assumed to be stationary and ergodic.The approach is based on hitting time decomposition.

Tail estimates for one-dimensional non-nearest-neighbor random walk in random environment

Suppose that the integers are assigned i.i.d. random variables {(β gx , . . . , β 1x , α x )} (each taking values in the unit interval and the sum of them being 1), which serve as an environment. This environment defines a random walk {X n } (called RWRE) which, when at x, moves one step of length 1 to the right with probability α x and one step of length k to the left with probability β kx for 1≤ k≤ g. For certain environment distributions, we determine the almost-sure asymptotic speed of the RWRE and show that the chance of the RWRE deviating below this speed has a polynomial rate of decay. This is the generalization of the results by Dembo, Peres and Zeitouni in 1996. In the proof we use a large deviation result for the product of random matrices and some tail estimates and moment estimates for the total population size in a multi-type branching process with random environment.

Scaling limit theorem for transient random walk in random environment

We construct a sequence of transient random walks in random environments and prove that by proper scaling, it converges to a diffusion process with drifted Brownian potential. To this end, we prove a counterpart of convergence for transient random walk in non-random environment, which is interesting itself.

Laws of iterated logarithm for transient random walks in random environments

We consider laws of iterated random walks in random environments. logarithm for one-dimensional transient A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environments, including independent identically distributed environments and uniformly ergodic environments.

Convergence of complex martingale for a branching random walk in an independent and identically distributed environment

We consider anR^(d)-valued discrete time branching random walk in an independent and identically distributed environment indexed by time n∈N.Let W_(n)(z)(z∈C^(d))be the natural complex martingale of the process.We show necessary and sufficient conditions for the L^(α)-convergence of W_(n)(z)forα>1,as well as its uniform convergence region.

ON THE RANGE OF RANDOM WALKS IN RANDOM ENVIRONMENT

The range of random walk on Z￣d in symmetric random environment is investigated. As results, it is proved that the strong law of large numbers for the range of random walk on Zdin some random environments holds if d≥ 3, and a weak law of large numbers holds for d= 1.