A quenched functional central limit theorem for planar random walks in random sceneries

Abstract

Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They were introduced at the end of the 70's by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field ξ = (ξ(x)) x∈Z d of i.i.d. random variables, which is called the random scenery, and a random walk S = (Sn) n∈N evolving in Z d , independent of the scenery. The RWRS Z = (Zn) n∈N is then defined as the accumulated scenery along the trajectory of the random walk, i.e., Zn := n k=1 ξ(S k ). The law of Z under the joint law of ξ and S is called "annealed", and the conditional law given ξ is called "quenched". Recently, functional central limit theorems under the quenched law were proved for Z by the first two authors for a class of transient random walks including walks with finite variance in dimension d ≥ 3. In this paper we extend their results to dimension d = 2.

Local Limit Theorem for Randomly Deforming Billiards

“…Proof of the Local Limit Theorem -Theorem 2.2. For every n ∈ N * , ℓ ∈ Z 2 and h ∈ B 1 , we set: (18) H ℓ,n h := P n 1 {Sn=ℓ} h .…”

“…Now let us prove the last point. For this, we use the general argument developed by Guillotin-Plantard, Dos Santos and Poisat in [18]. Indeed the proof of [18] only uses the following assumptions:…”

The quenched limiting distributions of a one-dimensional random walk in random scenery

“…Applying Lemma 3 (i) to the restriction of h on [−n, n], we may find a sequence λ j = λ j (ε, W, n) → ∞ such that sup |x|≤n |W λ j (x) − h(x)| ≤ ε. By applying (8) to f (x) = (W λ j (x) − h(x))1 (|x|≤n) , we have that…”

“…In [7], quenched central limit theorems (with the usual √ n-scaling and Gaussian law in the limit) were proved for a large class of transient random walks. More recently, in [8], the case of the planar random walk was studied, the authors proved a quenched version of the annealed central limit theorem obtained by Bolthausen in [2] In this note we consider the case of the simple symmetric random walk (S n ) n≥0 on Z, the random scenery (ξ x ) x∈Z is assumed to be centered with finite variance equal to one and there exists some δ > 0 such that E(|ξ 0 | 2+δ ) < ∞. We prove that under these assumptions, there is no quenched distributional limit theorem for K. In the sequel, for −∞ ≤ a < b ≤ ∞, we will denote by AC([a, b] → R) the set of absolutely continuous functions defined on the interval [a, b] with values in R. Recall that if f ∈ AC([a, b] → R), then the derivative of f (denoted byḟ ) exists almost everywhere and is Lebesgue integrable on [a, b].…”

“…Limit theorems for RWRS have a long history, we refer to [7] or [8] for a complete review. Distributional limit theorems for quenched sceneries (i.e.…”