Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes

1981 ◽  
Vol 18 (1) ◽  
pp. 122-130 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Processes ◽  
Existence Of Solutions ◽  
Strong Ergodicity ◽  
Exponential Ergodicity ◽  
Countable Space ◽  
Birth Death

For regular Markov processes on a countable space, we provide criteria for the forms of ergodicity in the title in terms of the existence of solutions to inequalities involving the Q-matrix of the process. An application to birth-death processes is given.

Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes

1981 ◽  
Vol 18 (01) ◽  
pp. 122-130 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Processes ◽  
Existence Of Solutions ◽  
Strong Ergodicity ◽  
Exponential Ergodicity ◽  
Countable Space ◽  
Birth Death

For regular Markov processes on a countable space, we provide criteria for the forms of ergodicity in the title in terms of the existence of solutions to inequalities involving the Q-matrix of the process. An application to birth-death processes is given.

Strong ergodicity for Markov processes by coupling methods

2002 ◽  
Vol 39 (4) ◽  
pp. 839-852 ◽  
Author(s):  
Yong-Hua Mao
Keyword(s):  
Markov Processes ◽  
Sufficient Conditions ◽  
Essential Spectra ◽  
Strong Ergodicity ◽  
One Dimensional ◽  
Coupling Methods ◽  
Explicit Criteria ◽  
Birth Death

In this paper, we apply coupling methods to study strong ergodicity for Markov processes, and sufficient conditions are presented in terms of the expectations of coupling times. In particular, explicit criteria are obtained for one-dimensional diffusions and birth-death processes to be strongly ergodic. As a by-product, strong ergodicity implies that the essential spectra of the generators for these processes are empty.

Strong ergodicity for Markov processes by coupling methods

2002 ◽  
Vol 39 (04) ◽  
pp. 839-852 ◽  
Author(s):  
Yong-Hua Mao
Keyword(s):  
Markov Processes ◽  
Sufficient Conditions ◽  
Essential Spectra ◽  
Strong Ergodicity ◽  
One Dimensional ◽  
Coupling Methods ◽  
Explicit Criteria ◽  
Birth Death

In this paper, we apply coupling methods to study strong ergodicity for Markov processes, and sufficient conditions are presented in terms of the expectations of coupling times. In particular, explicit criteria are obtained for one-dimensional diffusions and birth-death processes to be strongly ergodic. As a by-product, strong ergodicity implies that the essential spectra of the generators for these processes are empty.

scholarly journals Stochastic semigroups and their applications to biological models

2012 ◽  
Vol 45 (2) ◽  
Author(s):  
Katarzyna Pichór ◽  
Ryszard Rudnicki ◽  
Marta Tyran-Kamińska
Keyword(s):  
Markov Processes ◽  
Asymptotic Properties ◽  
Biological Models ◽  
Genes Expression ◽  
Piecewise Deterministic Markov Processes ◽  
Birth Death

AbstractSome recent results concerning generation and asymptotic properties of stochastic semigroups are presented. The general results are applied to biological models described by piecewise deterministic Markov processes: birth-death processes, the evolution of the genome, genes expression and physiologically structured models.

Proportional intensities and strong ergodicity for Markov processes

1983 ◽  
Vol 20 (01) ◽  
pp. 185-190 ◽  
Author(s):  
Mark Scott ◽  
Dean L. Isaacson
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Homogeneous Markov Process ◽  
Strong Ergodicity ◽  
Transition Functions ◽  
The Matrix ◽  
Intensity Functions ◽  
Time Point

By assuming the proportionality of the intensity functions at each time point for a continuous-time non-homogeneous Markov process, strong ergodicity for the process is determined through strong ergodicity of a related discrete-time Markov process. For processes having proportional intensities, strong ergodicity implies having the limiting matrix L satisfy L · P(s, t) = L, where P(s, t) is the matrix of transition functions.

Birth-death processes with an instantaneous reflection barrier

2003 ◽  
Vol 40 (01) ◽  
pp. 163-179 ◽  
Author(s):  
Anyue Chen ◽  
Kai Liu
Keyword(s):  
Markov Processes ◽  
Special Property ◽  
Uniqueness Criterion ◽  
Ergodic Processes ◽  
Equilibrium Distributions ◽  
Birth Death

A new structure with the special property that an instantaneous reflection barrier is imposed on the ordinary birth—death processes is considered. An easy-checking criterion for the existence of such Markov processes is first obtained. The uniqueness criterion is then established. In the nonunique case, all the honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. It is proved that honest processes are always ergodic without necessarily imposing any extra conditions. Equilibrium distributions for all these ergodic processes are established. Several examples are provided to illustrate our results.

Birth-death processes with disaster and instantaneous resurrection

2004 ◽  
Vol 36 (01) ◽  
pp. 267-292
Author(s):  
Anyue Chen ◽  
Hanjun Zhang ◽  
Kai Liu ◽  
Keith Rennolls
Keyword(s):  
Markov Processes ◽  
Special Property ◽  
Death Process ◽  
Mass Disaster ◽  
Uniqueness Criterion ◽  
Equilibrium Distributions ◽  
Existence Criteria ◽  
Birth Death

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results.

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