On random walks in random scenery

Relative Complexity of Random Walks in Random Scenery in the absence of a weak invariance principle for the local times

We answer the question of Aaronson about the relative complexity of Random Walks in Random Sceneries driven by either aperiodic two dimensional random walks, two-dimensional Simple Random walk, or by aperiodic random walks in the domain of attraction of the Cauchy distribution. A key step is proving that the range of the random walk satisfies the Fölner property almost surely.

Limit theorems for additive functionals of random walks in random scenery

We study the asymptotic behaviour of additive functionals of random walks in random scenery. We establish bounds for the moments of the local time of the Kesten and Spitzer process.These bounds combined with a previous moment convergence result (and an ergodicity result) imply the convergence in distribution of additive observables (with a normalization in n^(1/4)).When the sum of the observable is null, the previous limit vanishes and we prove the convergence in the sense of moments (with a normalization in n^(1/8)).

随机行走上的场景重建

Scenery Reconstruction for Random Walk on Random Scenery Systems

本文研究了在随机行走中面对对手篡改记录的情况下，如何仍能重建大部分场景的性质。

A limit theorem for a random walk in a stationary scenery coming from a hyperbolic dynamical system

In this paper, we extend a result of Kesten and Spitzer (1979). Let us consider a stationary sequence $(ξ\_k:=f(T^k(.)))\_k$ given by an invertible probability dynamical system and some centered function $f$. Let $(S\_n)\_n$ be a simple symmetric random walk on $Z$ independent of $(ξ\_k)\_k$. We give examples of partially hyperbolic dynamical systems and of functions $f$ such that $n^{-3/4}(ξ(S\_1)+...+ξ(S\_k))$ converges in distribution as $n$ goes to infinity.

Scenery entropy as an invariant of RWRS processes

Probabilistic models of random walks in random sceneries give rise to examples of probability-preserving dynamical systems. A point in the state spaces consists of a walk-trajectory and a scenery, and its `motion' corresponds to shifting the time-origin. These models were proposed as natural examples of non-Bernoulli K-automorphisms by Adler, Ornstein and Weiss. This was proved in a famous analysis by Kalikow using Ornstein's Very Weak Bernoulli characterization of Bernoulli processes. Since then, various authors have generalized this construction to give other examples, including some smooth …

Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts

Let $(X, \cal B, ν)$ be a probability space and let $Γ$ be a countable group of $ν$-preserving invertible maps of $X$ into itself. To a probability measure $μ$ on $Γ$ corresponds a random walk on $X$ with Markov operator $P$ given by $Pψ(x) = \sum_{a} ψ(ax) \, μ(a)$. A powerful tool is the spectral gap property for the operator $P$ when it holds. We consider various examples of ergodic $Γ$-actions and random walks and their extensions by a vector space: groups of automorphisms or affine transformations on compact nilmanifolds, random walk in random scenery on non amenable groups, translations …