Robust Weak Ergodicity And Stable Ergodicity

In this paper, we define robust weak ergodicity and study the relation between robust weak ergodicity and stable ergodicity for conservative partially hyperbolic systems. We prove that a C^r(r > 1) conservative partially hyperbolic diffeomorphism is stably ergodic if it is robustly weakly ergodic and has positive (or negative) central exponents on a positive measure set. Furthermore, if the condition of robust weak ergodicity is replaced by weak ergodicity, then the diffeomophism is an almost stably ergodic system. Additionally, we show in dimension three, a C^r(r > 1) conservative partially hyperbolic diffeomorphism can be approximated by stably ergodic systems if it is robustly weakly ergodic and robustly has non-zero central exponents.     

Dobrushin Coefficients of Ergodicity and Asymptotically Stable L 1-Contractions

We extend an inequality (which involves the Dobrushin coefficient of ergodicity; see Cohen et al.⁽⁴⁾) to any linear bounded operator with domain and codomain L ¹-spaces. We use the extended Dobrushin coefficient of ergodicity, that appears in the inequality, in order to obtain sufficient conditions for the uniform asymptotic stability of a positive contraction of an L ¹-space. We conclude the paper by studying a class of strongly asymptotically stable positive contractions.     

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.     

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     

Exponential and Strong Ergodicity for Markov Processes with an Application to Queues

For an ergodic continuous-time Markov process with a particular state in its space,the authors provide the necessary and sufficient conditions for exponential and strong ergodicity in terms of the moments of the first hitting time on the state.An application to the queue length process of M/G/1 queue with multiple vacations is given.     

Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents

This is a survey-type article whose goal is to review some recent results on existence of hyperbolic dynamical systems with discrete time on compact smooth manifolds and on coexistence of hyperbolic and non-hyperbolic behavior. It also discusses two approaches to the study of genericity of systems with nonzero Lyapunov exponents.      

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。     

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.     

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.     

Some New Results on Strong Ergodicity

Coupling method is used to obtain the explicit upper and lower bounds for convergence rates in strong ergodicity for Markov processes. For one-dimensional diffusion processes and birth-death processes, these bounds are sharp in the sense that the upper one and the lower one are only different by a constant. This announcement is an outline of an original research paper “Convergence Rates in Strong Ergodicity for Markov Processes” that will appear in Stoch. Process. Their Appl.     

马尔科夫过程的强遍历性和一致衰减性

本文证明了黎曼流形上的非爆炸正Harris常返扩散过程的强遍历性等价于某(任)一紧集击中时期望的一致有界性;而马尔科夫过程一致衰减当且仅当爆炸时的期望一致有界.     

随机环境下的MTAR模型的几何遍历性

在本文中,提出了随机环境下的MTAR模型的非常返性及其确定的导出序列几何遍历的几个充分条件.     

Ergodicity of a polling network with an infinite number of stations

A Markov polling system with infinitely many stations is studied. The topic is the ergodicity of the infinite-dimensional process of queue lengths. For the infinite-dimensional process, the usual type of ergodicity cannot prevail in general and we introduce a modified concept of ergodicity, namely, weak ergodicity. It means the convergence of finite-dimensional distributions of the process. We give necessary and sufficient conditions for weak ergodicity. Also, the “usual” ergodicity of the system is studied, as well as convergence of functionals which are continuous in some norm. This revised version was published online in June 2006 with corrections to the Cover Date.     

Ergodicity properties of rerouting strategies in queueing networks

In this paper we study ergodicity properties for simple Markovian models describing different rerouting policies for queueing systems with two Poisson arrival streams and two exponential servers. On their arrival, customers are either routed to their normal server or are rerouted to the alternate server. We model the extra work for rerouted customers by assuming that each rerouted customer generates several tasks for the alternate server. This model can become non-ergodic even when the total arrival rate is smaller than the total service rate in the system. We compare different strategies for rerouting customers on the basis of necessary and sufficient conditions for ergodicity. For this purpose we make use of Lyapunov functions.Senior Research Associate N.F.W.O. (Belgian National Foundation for Scientific Research). Part of the research leading to this paper was carried out within the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office of Science, Technology and Culture. The scientific responsibility rests with the authors.This author was partly supported by a Cátedra Patrimonial of the Mexican Council of Science and Technology.     

Ergodicity of Quasi-birth and Death Processes （Ⅰ）

Quasi-birth and death processes with block tridiagonal matrices find many applications in various areas. Neuts gave the necessary and sufficient conditions for the ordinary ergodicity and found an expression of the stationary distribution for a class of quasi-birth and death processes. In this paper we obtain the explicit necessary and sufficient conditions for/-ergodicity and geometric ergodicity for the class of quasi-birth and death processes, and prove that they are not strongly ergodic. Keywords ergodicity, quasi-birth and death process.