We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on $(0,+\infty)$ for such CB processes. We give a representation of the stationary measure in terms of scale functions of related Lévy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom distribution in the subcritical case. Finally we explore some further properties of the limit distributions.

This paper investigates tail asymptotics of stationary distributions and quasi-stationary distributions (QSDs) of continuous-time Markov chains on subsets of the non-negative integers. Based on the so-called flux-balance equation, we establish identities for stationary measures and QSDs, which we use to derive tail asymptotics. In particular, for continuous-time Markov chains with asymptotic power law transition rates, tail asymptotics for stationary distributions and QSDs are classified into three types using three easily computable parameters: (i) super-exponential distributions, (ii) exponential-tailed distributions, and (iii) sub-exponential distributions. Our approach to establish tail asymptotics of stationary distributions is different from the classical semimartingale approach, and we do not impose ergodicity or moment bound conditions. In particular, the results also hold for explosive Markov chains, for which multiple stationary distributions may exist. Furthermore, our results on tail asymptotics of QSDs seem new. We apply our results to biochemical reaction networks, a general single-cell stochastic gene expression model, an extended class of branching processes, and stochastic population processes with bursty reproduction, none of which are birth–death processes. Our approach, together with the identities, easily extends to discrete-time Markov chains.

We study stationary measures for iterated function systems (considered as random dynamical systems) consisting of two piecewise affine interval homeomorphisms, called Alsedà–Misiurewicz (AM) systems. We prove that for an open set of parameters, the unique non-atomic stationary measure for an AM system has Hausdorff dimension strictly smaller than $1$. In particular, we obtain singularity of these measures, answering partially a question of Alsedà and Misiurewicz [Random interval homeomorphisms. Publ. Mat. 58(suppl.) (2014), 15–36].

In this paper we prove a local exponential synchronization for Markovian random iterations of homeomorphisms of the circle $S^{1}$ , providing a new result on stochastic circle dynamics even for $C^1$ -diffeomorphisms. This result is obtained by combining an invariance principle for stationary random iterations of homeomorphisms of the circle with a Krylov–Bogolyubov-type result for homogeneous Markov chains.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a connected real semisimple Lie group, $V$ be a finite-dimensional representation of $G$ and $\mu $ be a probability measure on $G$ whose support spans a Zariski-dense subgroup. We prove that the set of ergodic $\mu $-stationary probability measures on the projective space $\mathbb{P}(V)$ is in one-to-one correspondence with the set of compact $G$-orbits in $\mathbb{P}(V)$. When $V$ is strongly irreducible, we prove the existence of limits for the empirical measures. We prove related results over local fields as the finiteness of the set of ergodic $\mu $-stationary measures on the flag variety of $G$.

We discuss various properties of Probabilistic Cellular Automata, suchas the structure of the set of stationary measures and multiplicity ofstationary measures (or phase transition) for reversible models.

In this paper, we investigate how fast the stationary distribution π(K) of an embedded Markov chain (time-stationary distribution q(K) of the GI/M/1/K queue converges to the stationary distribution π of the embedded Markov chain (time-stationary distribution q) of the GI/M/1 queue as K tends to infinity. Simonot (1997) proved certain equalities. We obtain sharper results than these by finding limit values limK→∞σ-K||π(K) - π|| and limK→∞σ-K||q(K) - q|| explicitly.

Let t ≥ 0 be a Markov transition probability semigroup on a general space satisfying a suitable φ-irreducibility condition. We show the existence of (i) a decay parameter λ ≥ 0 which is a common abscissa of convergence of the integrals ƒ estPt (x, A) dt for almost all x and all suitable A, (ii) a natural classification into λ-positive, λ-null and λ-transient cases. Moreover this classification is completely determined by any one of the h-skeleton chains of (Pt). We study the convergence of eλtPt(x, A) in the λ-positive case, and show that the limit f(X)π(A) (where f and π are the unique λ-invariant function and measure, normalized so that π(f) = 1) is reached at a uniform exponential rate of convergence, i.e. ||eλtPt(x, ·)-f(x)π(·)||f = O (e−αt) for some α > 0 and almost all x if there is a π-positive set such that the convergence is exponentially fast on this set. These results are used to deduce conditions for (Pt) to have quasi-stationary distributions.

This paper presents limit theorems for the population sizes of a Bienaymé–Galton–Watson process allowing immigration. For the non-critical cases it is known that the limit distribution is non-defective iff a logarithmic moment of the immigration distribution is finite. The new results of this paper are concerned with the situation where this moment is infinite and give limit theorems for a certain slowly varying function of the population size. A parallel discussion is given for the critical case and also for the continuous-time process.

The methods of the paper are used to give some results on the rate of decay of the transition probabilities and on the growth rate of the stationary measure. These in turn are used to obtain some limit theorems for a reversed-time process.

The aim of this paper is to present a comprehensive set of criteria for classifying as recurrent, transient, null or positive the sets visited by a general state space Markov chain. When the chain is irreducible in some sense, these then provide criteria for classifying the chain itself, provided the sets considered actually reflect the status of the chain as a whole. The first part of the paper is concerned with the connections between various definitions of recurrence, transience, nullity and positivity for sets and for irreducible chains; here we also elaborate the idea of status sets for irreducible chains. In the second part we give our criteria for classifying sets. When the state space is countable, our results for recurrence, transience and positivity reduce to the classical work of Foster (1953); for continuous-valued chains they extend results of Lamperti (1960), (1963); for general spaces the positivity and recurrence criteria strengthen those of Tweedie (1975b).

The information flow in discrete Markov systems provides a method for determining that such a system is ergodic. Estimates are obtained for the information flow in some classes of Markov systems and using these estimates criteria for the ergodicity of the systems are established.