We construct Cheeger-type bounds for the second eigenvalue of a substochastic transition probability matrix in terms of the Markov chain's conductance and metastability (and vice versa) with respect to its quasistationary distribution, extending classical results for stochastic transition matrices.

Consider a one-sided Markov additive process with an upper and a lower barrier, where each can be either reflecting or terminating. For both defective and nondefective processes, and all possible scenarios, we identify the corresponding potential measures, which help to generalize a number of results for one-sided Lévy processes. The resulting rather neat formulae have various applications in risk and queueing theories, and, in particular, they lead to quasistationary distributions of the corresponding processes.

In this paper we discuss the decay properties of Markov branching processes with disasters, including the decay parameter, invariant measures, and quasistationary distributions. After showing that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest, we obtain the exact value of the decay parameter λC. We show that the decay parameter can be easily expressed explicitly. We further show that the Markov branching process with disaster is always λC-positive. The invariant vectors, the invariant measures, and the quasidistributions are given explicitly.

We study the long-time behaviour of a Markov process evolving in N and conditioned not to hit 0. Assuming that the process comes back quickly from ∞, we prove that the process admits a unique quasistationary distribution (in particular, the distribution of the conditioned process admits a limit when time goes to ∞). Moreover, we prove that the distribution of the process converges exponentially fast in the total variation norm to its quasistationary distribution and we provide a bound for the rate of convergence. As a first application of our result, we bring a new insight on the speed of convergence to the quasistationary distribution for birth-and-death processes: we prove that starting from any initial distribution the conditional probability converges in law to a unique distribution ρ supported in N* if and only if the process has a unique quasistationary distribution. Moreover, ρ is this unique quasistationary distribution and the convergence is shown to be exponentially fast in the total variation norm. Also, considering the lack of results on quasistationary distributions for nonirreducible processes on countable spaces, we show, as a second application of our result, the existence and uniqueness of a quasistationary distribution for a class of possibly nonirreducible processes.

We investigate the distribution of the coalescence time (most recent common ancestor) for two individuals picked at random (uniformly) in the current generation of a continuous-time Bienaymé-Galton-Watson process founded t units of time ago. We also obtain limiting distributions as t → ∞ in the subcritical case. We extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.

We study the endemic behaviour of a homogeneously mixing SIS epidemic in a population of size N with a general infectious period, Q, by introducing a novel subcritical branching process with immigration approximation. This provides a simple but useful approximation of the quasistationary distribution of the SIS epidemic for finite N and the asymptotic Gaussian limit for the endemic equilibrium as N → ∞. A surprising observation is that the quasistationary distribution of the SIS epidemic model depends on Q only through

We introduce two stochastic chemostat models consisting of a coupled population-nutrient process reflecting the interaction between the nutrient and the bacteria in the chemostat with finite volume. The nutrient concentration evolves continuously but depends on the population size, while the population size is a birth-and-death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long-time behavior of the bacterial population conditioned to nonextinction. We prove the global existence of the process and its almost-sure extinction. The existence of quasistationary distributions is obtained based on a general fixed-point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.

We consider a birth–death process {X(t),t≥0} on the positive integers for which the origin is an absorbing state with birth coefficients λn,n≥0, and death coefficients μn,n≥0. If we define A=∑n=1∞ 1/λnπn and S=∑n=1∞ (1/λnπn)∑i=n+1∞ πi, where {πn,n≥1} are the potential coefficients, it is a well-known fact (see van Doorn (1991)) that if A=∞ and S<∞, then λC>0 and there is precisely one quasistationary distribution, namely, {aj(λC)}, where λC is the decay parameter of {X(t),t≥0} in C={1,2,...} and aj(x)≡μ1-1πjxQj(x), j=1,2,.... In this paper we prove that there is a unique quasistationary distribution that attracts all initial distributions supported in C, if and only if the birth–death process {X(t),t≥0} satisfies bothA=∞ and S<∞. That is, for any probability measure M={mi, i=1,2,...}, we have limt→∞ℙM(X(t)=j∣ T>t)= aj(λC), j=1,2,..., where T=inf{t≥0 : X(t)=0} is the extinction time of {X(t),t≥0} if and only if the birth–death process {X(t),t≥0} satisfies both A=∞ and S<∞.

We study quasistationary distributions on a drifted Brownian motion killed at 0, when +∞ is an entrance boundary and 0 is an exit boundary. We prove the existence of a unique quasistationary distribution and of the Yaglom limit.

Let {Mn}n≥0 be a nonnegative time-homogeneous Markov process. The quasistationary distributions referred to in this note are of the form QA(x) = limn→∞P(Mn ≤ x | M0 ≤ A, M1 ≤ A, …, Mn ≤ A). Suppose that M0 has distribution QA, and define TAQA = min{n | Mn > A, n ≥ 1}, the first time when Mn exceeds A. We provide sufficient conditions for QA(x) to be nonincreasing in A (for fixed x) and for TAQA to be stochastically nondecreasing in A.

Consider a continuous-time Markov process with transition rates matrix Q in the state space Λ ⋃ {0}. In the associated Fleming-Viot process N particles evolve independently in Λ with transition rates matrix Q until one of them attempts to jump to state 0. At this moment the particle jumps to one of the positions of the other particles, chosen uniformly at random. When Λ is finite, we show that the empirical distribution of the particles at a fixed time converges as N → ∞ to the distribution of a single particle at the same time conditioned on not touching {0}. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N → ∞ to the unique quasistationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations are of order 1 / N.

We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions of a Markovian bulk-arriving queue that stops immediately after hitting the zero state. Investigating such behavior is crucial in realizing the busy period and some other related properties of Markovian bulk-arriving queues. The exact value of the decay parameter λC is obtained and expressed explicitly. The invariant measures, invariant vectors, and quasistationary distributions are then presented. We show that there exists a family of invariant measures indexed by λ ∈ [0, λC]. We then show that, under some conditions, there exists a family of quasistationary distributions, also indexed by λ ∈ [0, λC]. The generating functions of these invariant measures and quasistationary distributions are presented. We further show that a stopped Markovian bulk-arriving queue is always λC-transient and some deep properties are revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

In this paper we study the busy period in GI/G/1 work-conserving queues. We give the exact asymptotics of the tail distribution of the busy period under the light tail assumptions. We also study the workload process in the M/G/1 system conditioned to stay positive.

The stochastic logistic model has been studied in various contexts, including epidemiology, population biology, chemistry and sociology. Among the model predictions, the quasistationary distribution and the mean time to extinction are of major interest for most applications, and a number of approximation formulae for these quantities have been derived. In this paper, previous approximation formulae are improved for two mathematically tractable cases: at the limit of the number of individuals N → ∞ (with relative error of the approximations of the order 𝒪(1/N)), and at the limit of the basic reproduction ratio R0 → ∞ (with relative error of the approximations of the order 𝒪(1/R0)). The mathematically rigorous formulae are then extended heuristically for other values of N and R0 > 1.