Quasi-stationarity and quasi-ergodicity of general Markov processes

Zhang, JunFei F.; Li, ShouMei M.; Song, Renming

doi:10.1007/s11425-014-4835-x

Cited by 34 publications

(33 citation statements)

References 22 publications

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“…Proof of Theorem 5. First we will show the exponential convergence towards the Qprocess essentially thanks to (18). In the second step, we will show the existence and uniqueness of the quasi-ergodic distribution using a method similar to that used in [8].…”

Section: Assumptions and General Resultsmentioning

confidence: 99%

“…We may extend (18) to general initial law µ and π : putting moreover 1/c 1 c 2 inside the constant, there exists C s,π > 0 only depending on s and π such that, for any s ≤ t ≤ u,…”

Section: Assumptions and General Resultsmentioning

confidence: 99%

“…In a same way, existence of quasi-ergodic distributions has been also shown for such processes. The reader can see [12,18,4] for the study on quasi-ergodic distributions in a very general framework. In this article, we will be interested in the existence of a Q-process, a quasi-limiting distribution and a quasi-ergodic distribution when (A t ) t∈I depends on the time.…”

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confidence: 99%

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Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries

Oçafrain

2020

Stochastic Processes and their Applications

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We investigate some asymptotic properties of general Markov processes conditioned not to be absorbed by the moving boundaries. We first give general criteria involving an exponential convergence towards the Q-process, that is the law of the considered Markov process conditioned never to reach the moving boundaries. This exponential convergence allows us to state the existence and uniqueness of the quasiergodic distribution considering either boundaries moving periodically or stabilizing boundaries. We also state the existence and uniqueness of a quasi-limiting distribution when absorbing boundaries stabilize. We finally deal with some examples such as diffusions which are coming down from infinity.

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Section: Assumptions and General Resultsmentioning

confidence: 99%

“…We may extend (18) to general initial law µ and π : putting moreover 1/c 1 c 2 inside the constant, there exists C s,π > 0 only depending on s and π such that, for any s ≤ t ≤ u,…”

Section: Assumptions and General Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

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Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries

Oçafrain

2020

Stochastic Processes and their Applications

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“…In particular, Yaglom [73] was the first to explicitly identify QS distributions for the subcritical Bienaymé-Galton-Watson branching process. Part of the results on QS distributions concern Markov chains on positive integers with an absorbing state at the origin [36,72,42,44,66,76]. Other objects of study are the extinction probabilities for continuous-time branching process and the Fleming-Viot process [1,43,59].…”

Section: Introductionmentioning

confidence: 99%

Yaglom limit for stable processes in cones

Bogdan¹,

Palmowski²,

Wang³

2018

Electron. J. Probab.

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We give the asymptotics of the tail of the distribution of the first exit time of the isotropic αstable Lévy process from the Lipschitz cone in R d . We obtain the Yaglom limit for the killed stable process in the cone. We construct and estimate entrance laws for the process from the vertex into the cone. For the symmetric Cauchy process and the positive half-line we give a spectral representation of the Yaglom limit.Our approach relies on the scalings of the stable process and the cone, which allow us to express the temporal asymptotics of the distribution of the process at infinity by means of the spatial asymptotics of harmonic functions of the process at the vertex; on the representation of the probability of survival of the process in the cone as a Green potential; and on the approximate factorization of the heat kernel of the cone, which secures compactness and yields a limiting (Yaglom) measure by means of Prokhorov's theorem.Keywords. Yaglom limit ⋆ stable process ⋆ Lipschitz cone ⋆ quasi-stationary measure ⋆ Green function ⋆ Martin kernel ⋆ excursions.

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“…Recently, many authors have extensively studied the quasi-ergodic distribution; see [5,11,19] for example. In existing research works, it often needs to assume the process is λ-positive (see, e.g., [2,5,8]), except some specific cases.…”

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confidence: 99%

On the quasi-ergodic distribution of absorbing Markov processes

Zhang

Zhu

2019

Statistics & Probability Letters

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In this paper, we give a sufficient condition for the existence of a quasi-ergodic distribution for absorbing Markov processes. Using an orthogonal-polynomial approach, we prove that the previous main result is valid for the birth-death process on the nonnegative integers with 0 an absorbing boundary and ∞ an entrance boundary. We also show that the quasi-ergodic distribution is stochastically larger than the unique quasi-stationary distribution in the sense of monotone likelihood-ratio ordering for the birth-death process. IntroductionLet (Ω, (F t ) t≥0 , (X t ) t≥0 , (P t ) t≥0 , (P x ) x∈E∪{∂} ) be a time-homogeneous Markov process with state space E ∪ {∂}, where (E, E) is a measurable space and ∂ ∈ E is a cemetery state. Let P x and E x stand for the probability and the expectation, respectively, associated with the process X when initiated from x. We assume that the process X has a finite lifetime T , i.e., for allx ∈ E,where T = inf{t ≥ 0 : X t = ∂}. We also assume that for all t ≥ 0 and x ∈ E, P x (t < T ) > 0.

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Quasi-stationarity and quasi-ergodicity of general Markov processes

Abstract: In this paper we give some general, but easy-to-check, conditions guaranteeing the quasistationarity and quasi-ergodicity of Markov processes. We also present several classes of Markov processes satisfying our conditions.

Cited by 34 publications

References 22 publications

Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries

Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries

Yaglom limit for stable processes in cones

On the quasi-ergodic distribution of absorbing Markov processes

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