By the technique of augmented truncations, we obtain the perturbation bounds on the distance of the finite-time state distributions of two continuous-time Markov chains (CTMCs) in a type of weaker norm than the V-norm. We derive the estimates for strongly and exponentially ergodic CTMCs. In particular, we apply these results to get the bounds for CTMCs satisfying Doeblin or stochastically monotone conditions. Some examples are presented to illustrate the limitation of the V-norm in perturbation analysis and to show the quality of the weak norm.

Inspired by the phase transition results for non-singular Gaussian actions introduced in [AIM19], we prove several phase transition results for non-singular Bernoulli actions. For generalized Bernoulli actions arising from groups acting on trees, we are able to give a very precise description of their ergodic-theoretical properties in terms of the Poincaré exponent of the group.

Let $X$ be a solenoid, i.e. a compact, finite-dimensional, connected abelian group with normalized Haar measure $\unicode[STIX]{x1D707}$, and let $\unicode[STIX]{x1D6E4}\rightarrow \operatorname{Aff}(X)$ be an action of a countable discrete group $\unicode[STIX]{x1D6E4}$ by continuous affine transformations of $X$. We show that the probability measure preserving action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ does not have the spectral gap property if and only if there exists a $p_{\text{a}}(\unicode[STIX]{x1D6E4})$-invariant proper subsolenoid $Y$ of $X$ such that the image of $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually solvable group, where $p_{\text{a}}:\operatorname{Aff}(X)\rightarrow \operatorname{Aut}(X)$ is the canonical projection. When $\unicode[STIX]{x1D6E4}$ is finitely generated or when $X$ is the $a$-adic solenoid for an integer $a\geq 1$, the subsolenoid $Y$ can be chosen so that the image $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually abelian group. In particular, an action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ by affine transformations on a solenoid $X$ has the spectral gap property if and only if $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is strongly ergodic.

The eigentime identity for one-dimensional diffusion processes on the halfline with an entrance boundary at ∞ is obtained by using the trace of the deviation kernel. For the case of an exit boundary at ∞, a similar eigentime identity is presented with the aid of the Green function. Explicit equivalent statements are also listed in terms of the strong ergodicity or the uniform decay for diffusion processes.

This work is concerned with a class of mean-field models given by a switching diffusion with a continuous-state-dependent switching process. Focusing on asymptotic properties, the regularity or nonexplosiveness, Feller continuity, and strong Feller continuity are established by means of introducing certain auxiliary processes and by making use of the truncations. Based on these results, exponential ergodicity is obtained under the Foster–Lyapunov drift conditions. By virtue of the coupling methods, the strong ergodicity or uniform ergodicity in the sense of convergence in the variation norm is established for the mean-field model with a Markovian switching process. Besides this, several examples are presented for demonstration and illustration.

In this paper, we investigate computable lower bounds for the best strongly ergodic rate of convergence of the transient probability distribution to the stationary distribution for stochastically monotone continuous-time Markov chains and reversible continuous-time Markov chains, using a drift function and the expectation of the first hitting time on some state. We apply these results to birth–death processes, branching processes and population processes.

A dual Markov branching process (DMBP) is by definition a Siegmund's predual of some Markov branching process (MBP). Such a process does exist and is uniquely determined by the so-called dual-branching property. Its q-matrix Q is derived and proved to be regular and monotone. Several equivalent definitions for a DMBP are given. The criteria for transience, positive recurrence, strong ergodicity, and the Feller property are established. The invariant distributions are given by a clear formulation with a geometric limit law.

In this paper we study polynomial and geometric (exponential) ergodicity for M/G/1-type Markov chains and Markov processes. First, practical criteria for M/G/1-type Markov chains are obtained by analyzing the generating function of the first return probability to level 0. Then the corresponding criteria for M/G/1-type Markov processes are given, using their h-approximation chains. Our method yields the radius of convergence of the generating function of the first return probability, which is very important in obtaining explicit bounds on geometric (exponential) convergence rates. Our results are illustrated, in the final section, in some examples.

The eigentime identity is proved for continuous-time reversible Markov chains with Markov generator L. When the essential spectrum is empty, let {0 = λ0 < λ1 ≤ λ2 ≤ ···} be the whole spectrum of L in L2. Then ∑ n≥1 λ n -1 < ∞ implies the existence of the spectral gap α of L in L∞. Explicit formulae are presented in the case of birth–death processes and from these formulae it is proved that ∑ n≥1 λ n -1 < ∞ if and only if α > 0.

This paper investigates the rate of convergence to the probability distribution of the embedded M/G/1 and GI/M/n queues. We introduce several types of ergodicity including l-ergodicity, geometric ergodicity, uniformly polynomial ergodicity and strong ergodicity. The usual method to prove ergodicity of a Markov chain is to check the existence of a Foster–Lyapunov function or a drift condition, while here we analyse the generating function of the first return probability directly and obtain practical criteria. Moreover, the method can be extended to M/G/1- and GI/M/1-type Markov chains.

This paper concentrates on investigating ergodicity and stability for generalised Markov branching processes with resurrection. Easy checking criteria including several clear-cut corollaries are established for ordinary and strong ergodicity of such processes. The equilibrium distribution is given in an elegant closed form for the ergodic case. The probabilistic interpretation of the results is clear and thus explained.

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.

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.

Characterizations of strong ergodicity for Markov chains using mean visit times have been found by several authors (Huang and Isaacson (1977), Isaacson and Arnold (1978)). In this paper a characterization of uniform strong ergodicity for a continuous-time non-homogeneous Markov chain is given. This extends the characterization, using mean visit times, that was given by Isaacson and Arnold.

The concept of strong ergodicity for discrete-time homogeneous Markov chains has been characterized in several ways (Dobrushin (1956), Lin (1975), Isaacson and Tweedie (1978)). In this paper the characterization using mean visit times (Huang and Isaacson (1977)) is extended to continuous-time Markov chains. From this it follows that for a certain subclass of continuous-time Markov chains, X(t), is strongly ergodic if and only if the associated embedded chain is Cesaro strongly ergodic.

For finite Markov chains, the concepts of ergodicity and strong ergodicity are equivalent, but this is not necessarily the case when the state space is infinite. In this note we give some new characterizations of strong ergodicity. These lead to simple necessary or sufficient criteria for strong ergodicity, which readily enable us to classify a number of examples.