Recurrence-transience criteria for storage processes

It is shown that general storage processes are classified into three types: positive recurrent, null recurrent, and transient processes. Sufficient conditions are given for transience and recurrence (null or positive recurrence), respectively. The sufficient condition for transience is shown to be necessary for transience in a special case (linear release rate case). This is a slight simplification of a known necessary and sufficient condition. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

On conservativeness and recurrence criteria for Markov processes

In the present paper, we give general criteria of conservativeness and recurrence for Markov processes associated with, not necessarily symmetric, Dirichlet spaces. The conservativeness criterion is applied to discuss a comparison theorem of conservativeness for diffusion processes. Also some sufficient conditions of conservativeness and recurrence for diffusion processes.are given.     

Analysis of a Recurrence Related to Critical Nonhomogeneous Branching Processes

Abstract

Some classes of controlled branching processes (with nonhomo-geneous migration or with nonhomo-geneous state-dependent immigration) lead in the critical case to a recurrence for the extinction probabilities. Under some additional conditions it is known that this recurrence depends on some parameter β and converges for 0 < β < 1. Now we show that the recurrence does converge for all positive values of the parameter β, which leads to an extension of some limit theorems for the corresponding branching processes. We also give a generalization of the recurrence and an asymptotic analysis of its behavior.     

Criteria on ergodicity and strong ergodicity of single death processes

Based on an explicit representation of moments of hitting times for single death processes, the criteria on ergodicity and strong ergodicity are obtained. These results can be applied for an extended class of branching processes. Meanwhile, some sufficient and necessary conditions for recurrence and exponential ergodicity as well as extinction probability for the processes are presented.     

Continuity of mean recurrence times in denumerable semi-Markov processes

Summary For a family of semi-Markov processes where the transition matrices for the embedded Markov chains and the mean sojourn times depend continuously on a parameter, we give equivalent as well as sufficient conditions for the continuity of the mean recurrence times. The results will be used in a subsequent paper on average costs in a dynamic programming model.This work was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 72 at the University of Bonn     

Strong ergodicity of the regime-switching diffusion processes

We provide criteria for the strong ergodicity of regime-switching diffusion processes. Our conditions are imposed on the coefficients of the processes. Particularly, we show that for regime-switching diffusions on the half line, if the corresponding diffusion on each fixed environment is strongly ergodic, then the regime-switching diffusion is strongly ergodic as well, which does not depend on the changing rate of the environment. Moreover, the converse is not always true, which is shown by an example. For transience, recurrence and positive recurrence, there is no such good consistency [R. Pinsky and M. Scheutzow, Some remarks and examples concerning the transience and recurrence of random diffusions, Ann. Inst. Henri. Poincaré 28 (1992) 519–536].     

Recurrence and transience for jump–diffusion processes

The purpose of this work is to obtain sufficient conditions for transience and recurrence of multidimensional jump–diffusion processes, which are driven by Brownian motion and Poisson random measure. The approach adopted here is to construct appropriate Lyapounov functions     

二次分支q—矩阵

R.R.Chen(1997)对连续时间分支过程进行了推广，其q-矩阵为（1．2），对这类q-矩阵，她得到了过程的唯一性准则，常返性和正常返性的条件。Zhang,Lin和Hou(1999)对转移函数的性质进行了讨论，得到了转移函数是随机单调的，强遍历，多项式一致收敛，指数遍历和Feller转移函数的条件，在R.R.Chen(1997)和Zhang,Lin和Hon(1999)的基础上本对二次分支过程进行了进一步的讨论，得到了过程的鞅性，同时还得到了大偏差上界估计成立的条件。     

Sporadic randomness,Maxwell's Demon and the Poincaré recurrence times

In the case of fully chaotic systems, the distribution of the Poincaré recurrence times is an exponential whose decay rate is the Kolmogorov–Sinai (KS) entropy. We address the discussion of the same problem, the connection between dynamics and thermodynamics, in the case of sporadic randomness, using the Manneville map as a prototype of this class of processes. We explore the possibility of relating the distribution of Poincaré recurrence times to “thermodynamics”, in the sense of the KS entropy, also in the case of an inverse power-law. This is the dynamic property that Zaslavsky [Physics Today 52 (8) (1999) 39] finds to be responsible for a striking deviation from ordinary statistical mechanics under the form of Maxwell's Demon effect. We show that this way of establishing a connection between thermodynamics and dynamics is valid only in the case of strong chaos, where both the sensitivity to initial conditions and the distribution of the Poincaré recurrence times are exponential. In the case of sporadic randomness, resulting at long times in the Lévy diffusion processes, the sensitivity to initial conditions is initially a power-law, but it becomes exponential again in the long-time scale, whereas the distribution of Poincaré recurrence times keeps, or gets, its inverse power-law nature forever, including the long-time scale where the sensitivity to initial condition becomes exponential. We show that a non-extensive version of thermodynamics would imply the Maxwell's Demon effect to be determined by memory, and thus to be temporary, in conflict with the dynamic approach to Lévy statistics. The adoption of heuristic arguments indicates that this effect is possible, as a form of genuine equilibrium, after completion of the process of memory erasure.     

Estimates of the Rate of Convergence of Runge–Kutta-Type Algorithms with Random Errors and with Consideration of the Calculation Time

We study conditions for the convergence of new types of algorithms, the recurrence algorithms with random calculation time and random errors, by using limit theorems for processes with semi-Markov switches. We prove theorems of the averaging-principle type. Concrete applications to numerical procedures of Runge–Kutta-type are obtained.     

Asymptotic Properties of Parabolic Systems for Null-Recurrent Switching Diffusions

This work is concerned with the asymptotic behavior of systems of parabolic equations arising fromnull-recurrent switching diffusions,which are diffusion processes modulated by continuous-time Markov chains.A sufficient condition for null recurrence is presented.Moreover,convergence rate of the solutions of systems ofhomogeneous parabolic equations under suitable conditions is established.Then a case study on verifying one ofthe conditions proposed is provided with the use of a two-state Markov chain.To verify the condition,boundaryvalue problems (BVPs) for parabolic systems are treated,which are not the usual two-point BVP type.Anextra condition in the interior is needed resulting in jump discontinuity of the derivative of the correspondingsolution.     

Bivariate step processes and random evolutions

A random evolution process constructed from regular step processes with a common state space and indexed on an evolution rule space is shown to be a regular step process on the product space. Conversely, it is shown that under mild conditions, any regular step process on a product space is equivalent to a random evolution process. Conditions are given on the cardinality of the spaces and on the parameters of the process that are sufficient for the process to have various recurrence and ergodicity properties. Applications to birth-death processes are given.     

Dynamical behavior of a hybrid switching SIS epidemic model with vaccination and Lévy jumps

In this paper, the dynamical behavior of a hybrid switching SIS epidemic model with vaccination and Lévy jumps is considered. Besides a standard geometric Brownian motion, another two driving processes are taken into account: a stationary Poisson point process and a continuous time finite-state Markov chain. Firstly, we establish sufficient conditions for persistence in the mean of the disease. Then we obtain sufficient conditions for extinction of the disease. In addition, we also establish sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching.     

广义生-灭过程

本文给出了具有突变率的广义生-灭过程的常返性、正常返性、指数遍历性及强 遍历性的充分必要条件.     

一般状态空间跳过程的Harris常返

研究了一般状态空间跳过程的Harris常返,利用马氏性,得到了跳过程Harris常返的几个等价条件.     

Stability and irreducibility of queueing systems with finite capacity

In this paper, we obtain a readily verifiable condition of stability for GI/G/1 queueing systems with finite capacity. A necessary and sufficient condition of irreducibility of the queueing size process is involved. Under this assumption, we derive general conditions of recurrence (positive recurrence) for the general process describing the state of the system. The conditions of irreducibility and recurrence are based on restrictions over the supports of the interarrival and the service distributions, which are easy to check in practice. The positive recurrence is also connected to the first moments of both distributions. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Dichotomy of Recurrence and Transience of Semi-Lévy Processes

A semi-Lévy process is an additive process with periodically stationary increments. In particular, it is a generalization of a Lévy process. The dichotomy of recurrence and transience of Lévy processes is well known, but this is not necessarily true for general additive processes. In this paper, we prove the recurrence and transience dichotomy of semi-Lévy processes. For the proof, we introduce a concept of semi-random walk and discuss its recurrence and transience properties. An example of semi-Lévy process constructed from two independent Lévy processes is investigated. Finally, we prove the laws of large numbers for semi-Lévy processes.     

Recurrence and Ergodicity of Switching Diffusions with Past-Dependent Switching Having a Countable State Space

This work focuses on recurrence and ergodicity of switching diffusions consisting of continuous and discrete components, in which the discrete component takes values in a countably infinite set and the rates of switching at current time depend on the value of the continuous component over an interval including certain past history. Sufficient conditions for recurrence and ergodicity are given. Moreover, the relationship between systems of partial differential equations and recurrence when the switching is past-independent is established under suitable conditions.     

Sur certaines suites récurrentes cubiques ayant deux ou trois termes nuls

This paper contributes to the classical problem of the determination of the zeros of a cubic recurrence. Some necessary conditions, dealing with the initial values of the recurrence and the roots of the associate polynomial, are given so that the recurrence has two or three zeros.     

The Weighted Transience and Recurrence of Markov Processes

Transience and recurrence are among the most important concepts in Markov processes. In this paper, we study the transience and recurrence for right processes with a given weight function, and characterize them by potentials, excessive functions, first hitting times and last exit times of the process. We also study the properties of recurrent states.