Jon. Aaronson, Benjamin Weiss
In this note we identify the distributional limits of non-negative, ergodic stationary processes, showing that all are possible. Consequences for infinite ergodic theory are also explored and new examples of distributionally stable- and $α$-rationally ergodic tranformations are presented.
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We prove distributional limit theorems for random walk adic transformations obtaining ergodic distributional limits of exponential chi squared form.
Non-stationary version of Ergodic Theorem for random dynamical systems
本文证明了非平稳随机动力系统的逐点遍历定理，并给出两个适用实例：非平稳迭代函数系统与非平稳随机矩阵乘积。
Ergodicity and weak mixing for group-indexed infinitely divisible stationary processes
研究群索引无限可分平稳过程，通过对概率可分平稳过程进行随机连续扩展构造，将问题简化，证明了遍历无限可分平稳过程是弱混合的，揭示遍历性蕴含弱混合性是无限可分性固有属性，去除群结构假设。
It follows from Oseledec Multiplicative Ergodic Theorem (or Kingmans Subadditional Ergodic Theorem) that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system f with exponential specification property and a Holder continuous matrix cocycle A, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of A displays distributional chaos of type 1.
Non-stationary Itô-Kawada and Ergodic Theorems for random isometries
本文研究紧空间上非平稳独立随机等距映射的遍历性质，在无确定性像真闭子集假设下证明弱-* 收敛、遍历定理与大偏差估计，并将其推广到紧群随机游走，得到非平稳 Itô-Kawada 定理。
We establish characterization results for the ergodicity of stationary symmetric $α$-stable (S$α$S) and $α$-Frechet random fields. We show that the result of Samorodnitsky [Ann. Probab. 33 (2005) 1782-1803] remains valid in the multiparameter setting, that is, a stationary S$α$S ($0<α<2$) random field is ergodic (or, equivalently, weakly mixing) if and only if it is generated by a null group action. Similar results are also established for max-stable random fields. The key ingredient is the adaption of a characterization of positive/null recurrence of group actions by Takahashi [Kodai Math. Se…
We revisit conservative/dissipative and positive/null decompositions of stationary max-stable processes. Originally, both decompositions were defined in an abstract way based on the underlying non-singular flow representation. We provide simple criteria which allow to tell whether a given spectral function belongs to the conservative/dissipative or positive/null part of the de Haan spectral representation. Specifically, we prove that a spectral function is null-recurrent iff it converges to $0$ in the Cesàro sense. For processes with locally bounded sample paths we show that a spectral functio…
We prove that the joint distribution of the occupation time ratios for ergodic transformations preserving an infinite measure converges to a multidimensional version of Lamperti's generalized arcsine distribution, in the sense of strong distributional convergence. Our results can be applied to interval maps and Markov chains. We adopt the double Laplace transform method, which has been utilized in the study of occupation times of diffusions on multiray. We also discuss the inverse problem.
This paper deals with measurable stationary symmetric stable random fields indexed by R^d and their relationship with the ergodic theory of nonsingular R^d-actions. Based on the phenomenal work of Rosinski(2000), we establish extensions of some structure results of stationary S alpha S processes to SαS fields. Depending on the ergodic theoretical nature of the underlying action, we observe different behaviors of the extremes of the field.
Functional limits for "tied down" occupation time processes of infinite ergodic transformations
该研究针对无限遍历变换的约束占有时间过程，证明了其泛函分布极限定理，明确极限为约束Mittag-Leffler过程，且相关变换的泛函约束更新混合性质强于已有研究[AS19]。
In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical "unbiased behavior" with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper we prove, using ergodic theory and the notion of category, that clustering (even very strong) is i…
Trimmed strong laws and distributional limits for exponentially mixing systems
本文研究了指数混合系统中截断遍历和的强律和分布极限，证明了α≥1时的截断大数定律，并在α>1/2时展示了非标准分布和正态分布的收敛性。
Stochastic bifurcation of a three-dimensional stochastic Kolmogorov system
本文研究三维随机 Kolmogorov 系统的随机分岔，依据噪声阈值刻画唯一遍历性、三类平稳测度及全局随机周期解，并证明小噪声极限。
We prove that any continuous and convex stationary ergodic Hamiltonian admits critical subsolutions, which are strict outside the random Aubry set. They make up, in addition, a dense subset of all critical subsolutions with respect to a suitable metric. If the Hamiltonian is additionally assumed of Tonelli type, then there exist strict subsolutions of class $\CC^{1,1}$ in $\R^N$. The proofs are based on the use of Lax--Oleinik semigroups and their regularizing properties in the stationary ergodic environment, as well as on a generalized notion of Aubry set.
{\abstract{\textwidth=4,5 in} A discrete time process, with law $μ$, is quasi-exchangeable if for any finite permutation $σ$ of time indices, the law $μ_σ$ of the resulting process is equivalent to $μ$. For a quasi-exchangeable stationary process we prove mainly (1) that if the process is ergodic then it is isomorphic to a Bernoulli process and (2) that if the family of all Radon-Nikodym derivatives $\{{dμ_σ\over dμ}\}$ is uniformly integrable then the process is a mixture of Bernoulli processes, which generalizes De Finetti's Theorem. We give application of (1) to some determinantal processes…
Tied-down occupation times of infinite ergodic transformations
该研究针对一类弱混合逐点对偶遍历变换，证明其游程后约束时刻停留时间的分布极限定理，提出约束更新混合性质以细化有理弱混合，并建立Gibbs-Markov与AFU映射的周期局部极限定理。
We define a random walk adic transformation associated to an aperiodic random walk on $G=\mathbb{Z}^{k}\times\mathbb{R}^{D-k}$ driven by a $β$-transformation and study its ergodic properties. In particular, this transformation is conservative, ergodic, infinite measure preserving and we prove that it is asymptotically distributionally stable and bounded rationally ergodic. Related earlier work appears in [AS] and [ANSS] for random walk adic transformations associated to an aperiodic random walk driven by a subshift of finite type.
Classifying Hyperbolic Ergodic Stationary Measures on Compact Complex Surfaces with Large Automorphism Groups
本文在不假设群含抛物元素的前提下，研究紧复曲面自同构群上有限支撑概率测度所生成的非初等群，并对双曲遍历 $μ$-平稳概率测度进行分类。
In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of $L^1(m)$ functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non-$L^1(m)$ functions. Here, we provide another distributional behavior of time averages of non-$L^1(m)$ functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered…
Motivated by a problem in ergodic Ramsey theory, Furstenberg and Katznelson introduced the notion of strong stationarity, showing that certain recurrence properties hold for arbitrary measure preserving systems if they are valid for strongly stationary ones. We construct some new examples and prove a structure theorem for strongly stationary systems. The building blocks are Bernoulli systems and rotations on nilmanifolds.
Parsings of Stationary Processes, Stopping Times and the Fundamental Pointwise Convergence Theorems of Ergodic Theory
本文引入平稳过程的单词集合解析概念及渐近分析框架，结合解析观察（基于定制停止时间的集合），从较弱的依概率收敛版本推导出逐点遍历定理与香农-麦克米兰-布莱曼定理。
This paper is inspired by the problem of understanding in a mathematical sense the Liouville quantum gravity on surfaces. Here we show how to define a stationary random metric on self-similar spaces which are the limit of nice finite graphs: these are the so-called hierarchical graphs. They possess a well-defined level structure and any level is built using a simple recursion. Stopping the construction at any finite level, we have a discrete random metric space when we set the edges to have random length (using a multiplicative cascade with fixed law $m$). We introduce a tool, the cut-off proc…
We study random perturbations of multidimensional piecewise expanding maps. We characterize absolutely continuous stationary measures (acsm) of randomly perturbed dynamical systems in terms of pseudo-orbits linking the ergodic components of absolutely invariant measures (acim) of the unperturbed system. We focus on those components, called least-elements, which attract pseudo-orbits. We show that each least element admits a neighbourhood which supports exactly one ergodic acsm of the random system. We use this result to identify random perturbations that exhibit a metastable behavior.
Invariant measures for reducible generalized Bratteli diagrams
本文研究可约广义布拉特利图上的概率遍历不变测度，证明经典平稳情形下特征值确定不变测度的结论在广义情形不成立，并刻画了含无穷多个简单标准子图或里程表子图的平稳与非平稳图类的不变测度集合及其支撑Vershik同胚的偏序条件。
Quasi-stationary and quasi-ergodic distributions in the Pelikan random map
研究可数状态空间上次随机离散时间马尔可夫链，以佩利肯动力系统变体为模型，产生无限多拟平稳分布，经数值模拟检验其稳定性，还确定了某些参数值下唯一拟遍历分布的存在性。
Motivated by the work of Busse et al. [6] on turbulent convection in a rotating layer, we exploit the long-run behavior for stochastic Lotka-Volterra (LV) systems both in pull-back trajectory and in stationary measure. It is proved stochastic decomposition formula describing the relation between solutions of stochastic and deterministic LV systems and stochastic Logistic equation. By virtue of this formula, it is verified that every pull-back omega limit set is an omega limit set for deterministic LV systems multiplied by the random equilibrium of the stochastic Logistic equation. This formula…
Assume that $T$ is a conservative ergodic measure preserving transformation of the infinite measure space $(X,\mathcal{A},μ)$.We study the asymptotic behaviour of occupation times of certain subsets of infinite measure. Specifically, we prove a Darling-Kac type distributional limit theorem for occupation times of barely infinite components which are separated from the rest of the space by a set of finite measure with c.f.-mixing return process. In the same setup we show that the ratios of occupation times of two components separated in this way diverge almost everywhere. These abstract results…
We consider stationary stochastic dynamical systems evolving on a compact metric space, by perturbing a deterministic dynamics with a random noise, added according to an arbitrary probabilistic distribution. We prove the maximal and pointwise ergodic theorems for the transfer operators associated to such systems. The results are extensions to noisy systems of some of the fundamental ergodic theorems for deterministic systems.
Stationary scattering problem (when the distance $r$ tends to infinity) and dynamical scattering problem (when the time $t$ tends to infinity) are considered for the 3D Schrödinger equation. A simple interconnection between the scattering amplitude (stationary case) and scattering operator (dynamical case) is given in the paper. This result is a quantum mechanical analog of the ergodic formulas in the classical mechanics.
Horizontally stationary generalized Bratteli diagrams
本文研究水平平稳 Bratteli 图，刻画其遍历尾不变概率测度，证明一类图中此类测度源于里程计测度扩张，并给出连续 Vershik 映射的存在条件。
We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner's seminal work.
Exact dimensionality of stationary measures for nonuniformly conformally contracting random diffeomorphisms
本文证明在单一负李雅普诺夫指数条件下，非均匀共形收缩随机微分同胚的平稳测度具有精确维数，其维数由Furstenberg熵与负指数之比给出，且无需离散性假设。
Even simply-defined, finite-state generators produce stochastic processes that require tracking an uncountable infinity of probabilistic features for optimal prediction. For processes generated by hidden Markov chains the consequences are dramatic. Their predictive models are generically infinite-state. And, until recently, one could determine neither their intrinsic randomness nor structural complexity. The prequel, though, introduced methods to accurately calculate the Shannon entropy rate (randomness) and to constructively determine their minimal (though, infinite) set of predictive feature…
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