A note on the central limit theorem for two-fold stochastic random walks in a random environment (original) (raw)
arXiv (Cornell University), 2017
We prove a central limit theorem under diffusive scaling for the displacement of a random walk on Z d in stationary and ergodic doubly stochastic random environment, under the H −1-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [10], where it is assumed that the stream tensor is in L max{2+δ,d} , with δ > 0. Our proof relies on an extension of the relaxed sector condition of [8], and is technically rather simpler than existing earlier proofs of similar results by Oelschläger [19] and Komorowski, Landim and Olla [10].
The Annals of Probability, 2017
We prove a central limit theorem under diffusive scaling for the displacement of a random walk on Z d in stationary and ergodic doubly stochastic random environment, under the H −1-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [Fluctuations in Markov Processes-Time Symmetry and Martingale Approximation (2012) Springer], where it is assumed that the stream tensor is in L max{2+δ,d} , with δ > 0. Our proof relies on an extension of the relaxed sector condition of [Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012) 463-476], and is technically rather simpler than existing earlier proofs of similar results by Oelschläger [Ann. Probab. 16 (1988) 1084-1126] and Komorowski, Landim and Olla [Fluctuations in Markov Processes-Time Symmetry and Martingale Approximation (2012) Springer].
Almost sure functional central limit theorem for ballistic random walk in random environment
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2009
We consider a non-nestling random walk in a product random environment. We assume an exponential moment for the step of the walk, uniformly in the environment. We prove an invariance principle (functional central limit theorem) under almost every environment for the centered and diffusively scaled walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively. Ω = P Z d , where P = {(p z ) z∈Z d : p z ≥ 0, z p z = 1} is the simplex of all probability vectors on Z d . Vector ω x = (ω x,z ) z∈Z d gives the transition probabilities out of state x, denoted by π x,y (ω) = ω x,y−x . To run the random walk, fix an environment ω and an initial state z ∈ Z d . The random walk X 0,∞ = (X n ) n≥0 in environment ω started at z is then the canonical Markov chain with state space Z d whose path measure P ω z satisfies P ω z (X 0 = z) = 1 and P ω z (X n+1 = y|X n = x) = π x,y (ω). On the space Ω we put its product σ-field S, natural shifts π x,y (T z ω) = π x+z,y+z (ω), and a {T z }-invariant probability measure P that makes the system (Ω, S, (T z ) z∈Z d , P) ergodic. In this paper P is an i.i.d. product measure on P Z d . In other words, the vectors (ω x ) x∈Z d are i.i.d. across the sites x under P. 1 2 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN
Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment
Probability Theory and Related Fields, 1997
We consider a model of random walk on Z m , m ! 2, in a dynamical random environment described by a ®eld n fn t x X tY x P Z m1 g. The random walk transition probabilities are taken as t1 yj t xY n t g 0 y À x y À xY gx. We assume that the variables fn t x X tY x P Z m1 g are i.i.d., that both 0 u and uY s are ®nite range in u, and that the random term uY Á is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for 0 , for all m ! 2. For m ! 3 there is a ®nite random (i.e., dependent on n) correction to the average of t , and there is a corresponding random correction of order O 1 t p to the C.L.T.. For m ! 5 there is a ®nite random correction to the covariance matrix of t and a corresponding correction of order O 1 t to the C.L.T.. Proofs are based on some new v p estimates for a class of functionals of the ®eld.
Almost sure functional central limit theorem for non-nestling random walk in random environment
2007
We consider a non-nestling random walk in a product random environment. We assume an exponential moment for the step of the walk, uniformly in the environment. We prove an invariance principle (functional central limit theorem) under almost every environment for the centered and diffusively scaled walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.
Quenched central limit theorem for random walks in doubly stochastic random environment
The Annals of Probability, 2018
We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the H −1-condition, with slightly stronger, L 2+ε (rather than L 2) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the nonreversible, divergence-free drift case, with unbounded (L 2+ε) stream tensor. This paper is a sequel of [Ann. Probab. 45 (2017) 4307-4347] and relies on technical results quoted from there.
Occupation time limit theorems for independent random walks
1994
We summarize many limit theorems for systems of independent simple random walks in Z. These theorems are classi ed into four classes: weak convergence, moderate deviations, large deviations and enormous deviations. A hierarchy of relations is pointed out and some open problems are posed. Extensions to function spaces are also mentioned.
Simple Transient Random Walks in One-dimensional Random Environment: the Central Limit Theorem
2006
We consider a simple random walk (dimension one, nearest neighbour jumps) in a quenched random environment. The goal of this work is to provide sufficient conditions, stated in terms of properties of the environment, under which the Central Limit Theorem (CLT) holds for the position of the walk. Verifying these conditions leads to a complete solution of the problem in the case of independent identically distributed environments as well as in the case of uniformly ergodic (and thus also weakly mixing) environments.