Klaus Schmidt
We survey distributional properties of $\mathbb{R}^d$-valued cocycles of finite measure preserving ergodic transformations (or, equivalently, of stationary random walks in $\mathbb{R}^d$) which determine recurrence or transience.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
After relating the notion of $ω$-recurrence in skew products to the range of values taken by partial ergodic sums and Lyapunov exponents, ergodic $\mathbb{Z}$-valued cocycles over an irrational rotation are presented in detail. First, the generic situation is studied and shown to be $1/n$-recurrent. It is then shown that for any $ω(n) <n^{-ε}$, where $ε>1/2$, there are uncountably many infinite staircases (a certain specific cocycle over a rotation) which are \textit{not} $ω$-recurrent, and therefore have positive Lyapunov exponent. A further section makes brief remarks regarding cocycles over…
Coexistence of two contrasting recurrence properties of certain non-integrable cocycles
本文研究了非积分协振子的递归性质，证明了此类系统在耗散的同时具有拓扑递归性。
We prove the annealed Central Limit Theorem for random walks in bistochastic random environments on $Z^d$ with zero local drift. The proof is based on a "dynamicist's interpretation" of the system, and requires a much weaker condition than the customary uniform ellipticity. Moreover, recurrence is derived for $d \le 2$.
We show that the "twisted" planar random walk - which results by summing up stationary increments rotated by multiples of a fixed angle - is recurrent under diverse assumptions on the increment process. For example, if the increment process is alpha-mixing and of finite second moment, then the twisted random walk is recurrent for every angle fixed choice of the angle out of a set of full Lebesgue measure, no matter how slow the mixing coefficients decay.
We discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular we show that a closed subgroup of a product of finitely many linear groups over local fields supports a recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces. The structure of closed subgroups of linear groups over local fields and the properties of group actions with respect to stationary me…
A Lindley process arises from classical studies in queueing theory and it usually reflects waiting times of customers in single server models. In this note we study recurrence of its higher dimensional counterpart under some mild assumptions on the tail behaviour of the underlying random walk. There are several links between the Lindley process and the associated random walk and we build upon such relations. We apply a method related to discrete subordination for random walks on the integer lattice together with various facts from the theory of fluctuations of random walks.
This paper has been withdrawn by the author due to a mistake in one of the main lemmas.
We study the recurrence properties of a random walk in a stratified medium.
In the present paper we find necessary and sufficient conditions for recurrence of random walks on arbitrary subgroups of the group of rational numbers $\mathbb{Q}$.
Considering homogeneous and oscillating random walks on the integers, we simplify classical works on recurrence of Spitzer and Kemperman, respectively. Some links with renewal theory are discussed.
The range process $R_n$ of a random walk is the collection of sites visited by the random walk up to time $n$. In this work we deal with the question of whether the range process of a random walk or the range process of a cocycle over an ergodic transformation is almost surely a Fölner sequence and show the following results: % (a) The size of the inner boundary $|\partial R_n|$ of the range of recurrent aperiodic random walks on $\mathbb{Z}^2$ with finite variance and aperiodic random walks in $\mathbb{Z}$ in the standard domain of attraction of the Cauchy distribution, divided by $\frac{n}{\…
We study the recurrence of inhomogeneous Markov chains in the plane, when the environment is horizontally stratified and the heterogeneity of quasi-periodic type.
We obtain various new limit theorems for random walks on SL_2(C) under low moment conditions. For non-elementary measures with a finite second moment, we prove a Local Limit Theorem for the norm cocycle, yielding the optimal version of a theorem of E. Le Page. For measures with a finite third moment, we obtain the Local Limit Theorem for the matrix coefficients, improving a recent result of Grama-Quint-Xiao and the authors, and Berry-Esseen bounds with optimal rate $O(1 / \sqrt n)$ for the norm cocycle and the matrix coefficients. The main tool is a detailed study of the spectral properties of…
Ergodicity of cocyles over 2-dimensional rotations
本文研究二维旋转上余循环的回复性与遍历性，考虑坏逼近无理数旋转及满足丢番图条件的不连续函数，主要关注ρ=2，d=1和2的情形。
We consider a random walk on a second countable locally compact topological space endowed with an invariant Radon measure. We show that if the walk is symmetric and if every subset which is invariant by the walk has zero or infinite measure, then one has escape of mass for almost every starting point. We then apply this result in the context of homogeneous random walks on infinite volume spaces, and deduce a converse to Eskin-Margulis recurrence theorem.
Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighborhood of the origin. We address this question in the case of some extended dynamical systems similar to planar random walks, including $\ZZ^2$-extension of hyperbolic dynamics. We define a pointwise recurrence rate and relate it to the dimension of the process, and establish a convergence in distribution of the rescaled return times near the origin.
Let $G$ be a semisimple Lie group acting on a space $X$, let $μ$ be a compactly supported measure on $G$, and let $A$ be a strongly irreducible linear cocycle over the action of $G$. We then have a random walk on $X$, and let $T$ be the associated shift map. We show that the cocycle $A$ over the action of $T$ is conjugate to a block conformal cocycle. This statement is used in the recent paper by Eskin-Mirzakhani on the classifications of invariant measures for the SL(2,R) action on moduli space. The ingredients of the proof are essentially contained in the papers of Guivarch and Raugi and als…
We study Markov chains on a lattice in a codimension-one stratified independent random environment, exploiting results established in [2]. First of all the random walk is transient in dimension at least three. Focusing on dimension two, both recurrence and transience can happen, but transience remains by far the most general situation. We identify the critical scale of the local drift along the strata corresponding to the frontier between the two regimes.
In this paper we show how polynomial walks can be used to establish a twisted recurrence for sets of positive density in $\mathbb{Z}^d$. In particular, we prove that if $Γ\leq \operatorname{GL}_d(\mathbb{Z})$ is finitely generated by unipotents and acts irreducibly on $\mathbb{R}^d$, then for any set $B \subset \mathbb{Z}^d$ of positive density, there exists $k \geq 1$ such that for any $v \in k \mathbb{Z}^d$ one can find $γ\in Γ$ with $γv \in B - B$. Our method does not require the linearity of the action, and we prove a twisted recurrence for semigroups of maps from $\mathbb{Z}^d$ to $\mathb…
Oscillating and nonsummable Radon-Nikodym cocycles along the forward geodesic of measure-class-preserving transformations
本研究通过构造Cantor空间上的乘积测度，得到沿最小删除映射前向测地线振荡或不可和趋于零的Radon-Nikodym上循环，解答了相关公开问题，并借助随机游走与Chung-Fuchs定理分析振荡实例。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。