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We introduce and study the basic properties of two ergodic stochastic control problems associated with the quasistationary distribution (QSD) of a diffusion process X relative to a bounded domain. The two problems are in some sense dual, with one defined in terms of the generator associated with X and the other in terms of its adjoint. Besides proving wellposedness of the associated Hamilton–Jacobi–Bellman equations, we describe how they can be used to characterize important properties of the QSD. Of particular note is that the QSD itself can be identified, up to normalization, in terms of the cost potential of the control problem associated with the adjoint. [ABSTRACT FROM AUTHOR]

A Quasi-Stationary Distribution (QSD) for a Markov process with an almost surely hit absorbing state is a time-invariant initial distribution for the process conditioned on not being absorbed by any given time. An initial distribution for the process is in the domain of attraction of some QSD ν if the distribution of the process a time t, conditioned not to be absorbed by time t converges to ν as t tends to infinity. We study Brownian motion with constant drift on the half line [0, ∞) absorbed at 0. Previous work by Martínez and San Martín (1994) and Martinez et al. (1998) identifies all QSDs and provides a nearly complete characterization for their domains of attraction. Specifically, it was shown that if the distribution a well-defined exponential tail (including the case of lighter than any exponential tail), then it is in the domain of attraction of a QSD determined by the exponent. In this work we expand the discussion regarding the dependence on the initial distribution through (1) Obtaining a new approach to existing results, explaining the direct relation between a QSD and an initial distribution in its domain of attraction; and (2) Considering a wide class of heavy-tailed initial distributions, where non-trivial limits are obtained under appropriate scaling. [ABSTRACT FROM AUTHOR]

A periodically driven, moderately anharmonic oscillator constitutes an ideal model system for investigating quantum resonances, which are amenable to a quantum pendulum approximation. In the present paper, I study the quasi-stationary Floquet-state occupation probabilities which emerge when such a resonantly driven system is coupled to a heat bath. It is demonstrated that the Floquet state which is associated with the ground state of the pendulum turns into an effective ground state, carrying the highest population in the strong-driving regime. Moreover, the population of this effective Floquet ground state can even exceed that of the undriven system's true ground state at the same bath temperature. These effects can be optimized by suitably engineering the properties of the bath. [ABSTRACT FROM AUTHOR]

The average run length (ARL), with its various phenotypes, is the prevailing performance measure for evaluating control charts, or change-point detection schemes. Essentially, the ARL counts the number of observations until the corresponding procedure flags a change. To enable a fair comparison between competing designs, one frequently deploys the steady-state ARL. Differing from the older concept of the zero-state ARL (which assumes that the to-be-detected change occurs immediately at startup or never), the former measure postulates this change's appearance after reaching some steady state. Considering different notions (primarily conditional and cyclical ones) of the measure, we recapitulate its historical development; provide a critical discussion of its often-careless exploitation, including a few misconceptions; and derive some new mathematical characterizations that permit its easy calculation. [ABSTRACT FROM AUTHOR]

Abstract In the present work we characterize the existence of quasistationary distributions for diffusions on (0 , ∞) allowing singular behavior at 0 and ∞. If absorption at 0 is certain, we show that there exists a quasistationary distribution as soon as the spectrum of the generator is strictly positive. This complements results of Cattiaux et al. (2009) and Kolb and Steinsaltz (2012) for 0 being a regular boundary point and extends results by Cattiaux et al. (2009) on singular diffusions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

Conditioning a multitype Galton-Watson process to stay alive into the indefinite future leads to what is known as its associated Q-process. We show that the same holds true if the process is conditioned to reach a positive threshold or a non-absorbing state. We also demonstrate that the stationary measure of the Q-process, obtained by construction as two successive limits (first by delaying the extinction in the original process and next by considering the long-time behavior of the obtained Q-process), is as a matter of fact a double limit. Finally, we prove that conditioning a multitype branching process on having an infinite total progeny leads to a process presenting the features of a Q-process. It does not however coincide with the original associated Q-process, except in the critical regime. [ABSTRACT FROM AUTHOR]

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 Levy 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. [ABSTRACT FROM AUTHOR]

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 lawto 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: We consider multidimensional discrete valued random walks with nonzero drift killed when leaving general cones of the euclidean space. We find the asymptotics for the exit time from the cone and study weak convergence of the process conditioned on not leaving the cone. We get quasistationarity of its limiting distribution. Finally we construct a version of the random walk conditioned to never leave the cone. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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 → α. A surprising observation is that the quasistationary distribution of the SIS epidemic model depends on Q only through E[Q] [ABSTRACT FROM AUTHOR]

Abstract: Methods for predicting the probability and timing of a species’ extinction are typically based on single species population dynamics. Assessments of extinction risk often lack effects of interspecific interactions. We study a birth and death process in which the death rate includes an effect of predation. Predation is included via a general nonlinear expression for the functional response of predation to prey density. We investigate the effects of the foraging parameters (e.g. attack rate and handling time) on the mean time to extinction. Mean time to extinction varies by orders of magnitude when we alter the foraging parameters, even when we exclude the effects of these parameters on the equilibrium population size. Conclusions are robust to assumptions about initial conditions and variable predator abundance. These findings clearly show that accounting for the nature of interspecific interactions is likely to be critically important when estimating extinction risk. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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 ... and ..., where {πn, n ≥ 1} are the potential coefficients, it is a well-known fact (See van Doom (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 = {l, 2, ...} and aj(x) ≡ μ1-1xπjxQj(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 both A = ∞ 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 < ∞. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., the first time when Mn exceeds A. We provide sufficient conditions for QA (x) to be nonincreasing in A (for fixed x) and for Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. to be stochastically nondecreasing in A. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 λ C-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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]