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.      

Random walk on the random connection model

We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices $x$ and $y$ are connected with probability that asymptotically behaves like ${|x?y|}^{?\alpha }$ with $\alpha >d$, where $d$ denotes the dimension of the underlying Euclidean space. More precisely, focus is on the random connection model in which the vertex set is given by the realization of a homogeneous Poisson point process. We show that this random graph exhibits similar properties as classical discrete long-range percolation models studied by Berger (2002) with regard to recurrence and transience of the random walk. Moreover, we address a question which is related to a conjecture by Heydenreich, Hulshof and Jorritsma (2017) for this graph.     

Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment

Summary We consider a model of random walk on ℤ^ν, ν≥2, in a dynamical random environment described by a field ξ={ξ _t (x): (t,x)∈ℤ^ν+1}. The random walk transition probabilities are taken as P(X _{t +1}= y|X _t = x,ξ _t =η) =P ₀( y−x)+ c(y−x;η(x)). We assume that the variables {ξ _t (x):(t,x) ∈ℤ^ν+1} are i.i.d., that both P ₀(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P ₀, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X _t , and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X _t and a corresponding correction of order to the C.L.T.. Proofs are based on some new L ^p estimates for a class of functionals of the field. Received: 4 January 1996/In revised form: 26 May 1997     

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.     

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.     

On competitive Lotka-Volterra model in random environments

Focusing on competitive Lotka-Volterra model in random environments, this paper uses regime-switching diffusions to model the dynamics of the population sizes of n different species in an ecosystem subject to the random changes of the external environment. It is demonstrated that the growth rates of the population sizes of the species are bounded above. Moreover, certain long-run-average limits of the solution are examined from several angles. A partial stochastic principle of competitive exclusion is also derived. Finally, simple examples are used to demonstrate our findings.     

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.     

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.     

Multiscale analysis of exit distributions for random walks in random environments

We present a multiscale analysis for the exit measures from large balls in , of random walks in certain i.i.d. random environments which are small perturbations of the fixed environment corresponding to simple random walk. Our main assumption is an isotropy assumption on the law of the environment, introduced by Bricmont and Kupiainen. Under this assumption, we prove that the exit measure of the random walk in a random environment from a large ball, approaches the exit measure of a simple random walk from the same ball, in the sense that the variational distance between smoothed versions of these measures converges to zero. We also prove the transience of the random walk in random environment. The analysis is based on propagating estimates on the variational distance between the exit measure of the random walk in random environment and that of simple random walk, in addition to estimates on the variational distance between smoothed versions of these quantities. Partially supported by NSF grant DMS-0503775.     

An iterative approximation procedure for the distribution of the maximum of a random walk

Let I(F) be the distribution function (d.f.) of the maximum of a random walk whose i.i.d. increments have the common d.f. F and a negative mean. We derive a recursive sequence of embedded random walks whose underlying d.f.'s F_k converge to the d.f. of the first ladder variable and satisfy FF₁F₂ on [0,∞) and I(F)=I(F₁)=I(F₂)=. Using these random walks we obtain improved upper bounds for the difference of I(F) and the d.f. of the maximum of the random walk after finitely many steps.     

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.     

Asymptotic direction for random walks in random environments

In this paper we study the existence of an asymptotic direction for random walks in random i.i.d. environments (RWRE). We prove that if the set of directions where the walk is transient contains a non-empty open set, the walk admits an asymptotic direction. The main tool to obtain this result is the construction of a renewal structure with cones. We also prove that RWRE admits at most two opposite asymptotic directions.     

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.     

Three-term asymptotic expansions for the moments of the random walk with triangular distributed interference of chance

In this study, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of ergodic distribution of the process X(t) are obtained when the random variable ζ₁, which is describing a discrete interference of chance, has a triangular distribution in the interval [s, S] with center (S + s)/2. Based on these results, the asymptotic expansions with three-term are obtained for the first four moments of the ergodic distribution of X(t), as a ≡ (S − s)/2 → ∞. Furthermore, the asymptotic expansions for the variance, skewness and kurtosis of the ergodic distribution of the process X(t) are established. Finally, by using Monte Carlo experiments it is shown that the given approximating formulas provide high accuracy even for small values of parameter a.     

Random walks in a strongly sparse random environment

The integer points (sites) of the real line are marked by the positions of a standard random walk with positive integer jumps. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the random walk are supported by a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity and assuming additionally that the distribution tail of the jumps is regularly varying at infinity we consider a nearest neighbor random walk on the set of integers having jumps $\pm 1$ with probability $1∕2$ at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in Buraczewski et al. (2019) for the case of moderate sparsity and in Matzavinos et al. (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable.     

Laws of iterated logarithm for transient random walks in random environments

We consider laws of iterated logarithm for one-dimensional transient random walks in random environments. 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.     

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)").     

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.