One-sided asymptotically mean stationary channels

This paper proposes an analysis of asymptotically mean stationary (AMS) communication channels. A hierarchy based on stability properties (stationarity, quasi-stationarity, recurrence and asymptotically mean stationarity) of channels is identified. Stationary channels are a subclass of quasi-stationary channels which are a subclass of recurrent AMS channels which are a subclass of AMS channels. These classes are proved to be stable under Markovian composition of channels (e.g., the cascade of AMS channels is an AMS channel). Characterizations of channels of each class are given. Some properties of the quasi-stationary mean of a channel are established. Finally, ergodicity conditions of AMS channels are gathered.     

复平稳随机序列均值遍历性的两个充要条件

遍历性是平稳随机过程的一个核心问题,在实践中有重要的应用.这里首先给出复平稳序列均值遍历性的一个等价定义,在此基础上证明了复平稳序列情形下均值遍历性的两个充要条件,并给出一个形式简单的推论,这些结果有助于深入的理解复平稳过程的均值遍历性,同时也为估计复平稳过程的均值提供了一种简单高效的方法.     

齐次马氏链强弱遍历等价关系的一种证明

主要给出齐次马氏链强遍历和弱遍历等价关系的一个直接证明.     

Shift ergodicity for stationary Markov processes

In this paper shift ergodicity and related topics are studied for certain stationary processes. We first present a simple proof of the conclusion that every stationary Markov process is a generalized convex combination of stationary ergodic Markov processes. A direct consequence is that a stationary distribution of a Markov process is extremal if and only if the corresponding stationary Markov process is time ergodic and every stationary distribution is a generalized convex combination of such extremal ones. We then consider space ergodicity for spin flip particle systems. We prove space shift ergodicity and mixing for certain extremal invariant measures for a class of spin systems, in which most of the typical models, such as the Voter Models and the Contact Models, are included. As a consequence of these results we see that for such systems, under each of those extremal invariant measures, the space and time means of an observable coincide, an important phenomenon in statistical physics. Our results provide partial answers to certain interesting problems in spin systems.     

On a class of asymptotically stationary harmonizable processes

We prove that every harmonizable process with σ-finite bimeasure is asymptotically stationary and we give its associated spectral measure.     

Ergodicity of Particle Systems

We study shift ergodicity, mixing, and related problems for invariant measures of interacting particle systems. The models we consider here include ferromagnetic stochastic Ising models, voter models, contact processes, exclusion processes, three-opinion noisy biased voter models, multi-opinion voter models, etc. Our results answer some questions for these models. One of the main techniques involved is a duality argument.     

Remark on the Abel Ergodicity of Quasi-pseudo-resolvents

In this note, some results concerning Abel ergodicity of quasi-pseudo-resolvents are obtained.AMS Subject Classification (1991): 47D05.Innovation Project for University Prominent Research Talents of Henan, China     

A construction of singular overlapping asymmetric self-similar measures

Summary In [8] we found a class of overlapping asymmetric self-similar measures on the real line, which are generically absolutely continuous with respect to the Lebesgue measure. Here we construct exceptional measures in this class being singular.     

Singular stationary measures are not always fractal

We know of few explicit results to insure that stationary measures are simultaneously (i) singular, (ii) nonatomic, (iii) with interval support, and (iv) unique. Such results would appear useful, to further separate the analytic notion of singular from the geometric notion of fractal. We prove two general theorems, one for maps of [0,1] into [0, 1], the other for 2×2 random matrices. In each setting, we study measures supported on two points of the transformation space, and we provide sufficient conditions to insure that the stationary measures satisfy (i)–(iv).     

Ergodicity of a polling network

The polling network considered here consists of a finite collection of stations visited successively by a single server who is following a Markovian routing scheme. At every visit of a station a positive random number of the customers present at the start of the visit are served, whereupon the server takes a positive random time to walk to the station to be visited next. The network receives arrivals of customer groups at Poisson instants, and all customers wait until served, whereupon they depart from the network. Necessary and sufficient conditions are derived for the server to be able to cope with the traffic. For the proof a multidimensional imbedded Markov chain is studied.     

Ergodicity of one-dimensional regime-switching diffusion processes

In this work, for a one-dimensional regime-switching diffusion process, we show that when it is positive recurrent, then there exists a stationary distribution, and when it is null recurrent, then there exists an invariant measure. We also provide the explicit representation of the stationary distribution and invariant measure based on the hitting times of the process.     

几类遍历性的显式判别准则

关于一维扩散过程或生灭过程的几类遍历性,新近很快地找到了显式判别准则,详见表5．1和5．2,其一是生灭过程的指数遍历性判准,曾是马链的一个长期未解问题,有趣的是,此论题与多个数学分支有关,我们详细介绍后一问题的想法,也建议了若干进一步研究课题。     

On Quantum Unique Ergodicity for Locally Symmetric Spaces

We construct an equivariant microlocal lift for locally symmetric spaces. In other words, we demonstrate how to lift, in a semi-canonical fashion, limits of eigenfunction measures on locally symmetric spaces to Cartan-invariant measures on an appropriate bundle. The construction uses elementary features of the representation theory of semisimple real Lie groups, and can be considered a generalization of Zelditch’s results from the upper half-plane to all locally symmetric spaces of noncompact type. This will be applied in a sequel to settle a version of the quantum unique ergodicity problem on certain locally symmetric spaces. The second author was supported in part by NSF Grant DMS-0245606. Part of this work was performed at the Clay Institute Mathematics Summer School in Toronto. Received: September 2005 Revision: August 2006 Accepted: August 2006     

Ergodicity of nonlinear first order autoregressive models

Criteria are derived for ergodicity and geometric ergodicity of Markov processes satisfyingX _n+1 =f(X _n )+(X _n ) _n+1 , wheref, are measurable, { _n } are i.i.d. with a (common) positive density,E| _n |>. In the special casef(x)/x has limits, , asx– andx+, respectively, it is shown that <1, <1, <1 is sufficient for geometric ergodicity, and that <-1, 1, 1 is necessary for recurrence.     

中心指数约束的部分双曲系统的遍历性

对于给定紧光滑黎曼流形M上的一个C~2保体积部分双曲微分同胚f,若对所有不变测度的中心方向的指数都满足中心指数约束条件,则f本质可达蕴含遍历.特别地,若对所有不变测度,中心方向指数都为零,则f本质可达蕴含遍历.这一结果部分回答了Pesin提出的两个公开问题.     

Time reversal and stationary Gibbs measures

Markov chains on an infinite product space are considered whose transition kernel is of the Gibbsian type. It is proved that then a stationary probability measure is Gibbsian if and only if the transition kernel of the reversed chain is also Gibbsian.     

带随机延滞的ARCH模型的几何遍历性

本文对带随机延滞的ARCH模型进行了分析,并得到该模型伴随几何遍历的一个判别准则.     

Uniform Exponential Ergodicity of Stochastic Dissipative Systems

We study ergodic properties of stochastic dissipative systems with additive noise. We show that the system is uniformly exponentially ergodic provided the growth of nonlinearity at infinity is faster than linear. The abstract result is applied to the stochastic reaction diffusion equation in ^d with d3.     

Implicit functions and sensitivity of stationary points

We consider the spaceL(D) consisting of Lipschitz continuous mappings fromD to the Euclideann-space ⁿ ,D being an open bounded subset of ⁿ . LetF belong toL(D) and suppose that solves the equationF(x) = 0. In case that the generalized Jacobian ofF at is nonsingular (in the sense of Clarke, 1983), we show that forG nearF (with respect to a natural norm) the systemG(x) = 0 has a unique solution, sayx(G), in a neighborhood of Moreover, the mapping which sendsG tox(G) is shown to be Lipschitz continuous. The latter result is connected with the sensitivity of strongly stable stationary points in the sense of Kojima (1980); here, the linear independence constraint qualification is assumed to be satisfied.