On the quasi-ergodic distribution of absorbing Markov processes

He, Guoman; Zhang, Hanjun; Zhu, Yu

doi:10.1016/j.spl.2019.02.001

Cited by 16 publications

(19 citation statements)

References 23 publications

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“…Analogously, ergodicity is a potentially problematic concept in this context. Nonetheless, given a unique quasi-stationary distribution for a Markov process (X t ) t≥0 on a state space E, one can derive the existence of a quasi-ergodic distribution m [3], characterized by Building on recent results by Villemonais, Champagnat, He and others [6,5,12], we obtain as our main result the existence of a conditioned Lyapunov exponent:…”

Section: Introductionmentioning

confidence: 82%

“…The existence of the QED m with the characterization of η as given in (b) follows from [12], which combines results from [6] and [3].…”

Section: Example: Local Pitchfork Bifurcation With Additive Noisementioning

confidence: 95%

“…These time averages conditioned on survival converge to an integral with respect to the relevant quasi-ergodic distribution. Existence of the quasi-ergodic distribution follows from standard theory [5,8,12] when d = 1 and from an assumption similar to [6, Assumption (A')] when d ≥ 2.…”

Section: )mentioning

confidence: 99%

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Conditioned Lyapunov exponents for random dynamical systems

Engel

Lamb

Rasmussen

2019

Trans. Amer. Math. Soc.

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We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to trajectories that stay within a bounded domain for asymptotically long times. This is motivated by the desire to characterize local dynamical properties in the presence of unbounded noise (when almost all trajectories are unbounded). We illustrate its use in the analysis of local bifurcations in this context.The theory of conditioned Lyapunov exponents of stochastic differential equations builds on the stochastic analysis of quasi-stationary distributions for killed processes and associated quasi-ergodic distributions. We show that conditioned Lyapunov exponents describe the local stability behaviour of trajectories that remain within a bounded domain and -in particular -that negative conditioned Lyapunov exponents imply local synchronisation. Furthermore, a conditioned dichotomy spectrum is introduced and its main characteristics are established. 1 We note that random dynamical systems with bounded noise suffer less from this problem, but such systems are generally not amenable to techniques from stochastic analysis and we will not consider these here. We refer to [2,13,14,19] for alternative approaches to bifurcations of random dynamical systems in the bounded noise context.

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Section: Introductionmentioning

confidence: 82%

“…The existence of the QED m with the characterization of η as given in (b) follows from [12], which combines results from [6] and [3].…”

Section: Example: Local Pitchfork Bifurcation With Additive Noisementioning

confidence: 95%

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Conditioned Lyapunov exponents for random dynamical systems

Engel

Lamb

Rasmussen

2019

Trans. Amer. Math. Soc.

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“…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.…”

mentioning

confidence: 99%

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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“…2.1] by providing the convergence estimate in 1/T . The interested reader might look into [6] for nice domination properties between the quasi-stationary distribution α and the probability β.…”

Section: Introductionmentioning

confidence: 99%

Uniform convergence to the $Q$-process

Champagnat¹,

Villemonais²

2017

Electron. Commun. Probab.

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The first aim of the present note is to quantify the speed of convergence of a conditioned process toward its Q-process under suitable assumptions on the quasi-stationary distribution of the process. Conversely, we prove that, if a conditioned process converges uniformly to a conservative Markov process which is itself ergodic, then it admits a unique quasi-stationary distribution and converges toward it exponentially fast, uniformly in its initial distribution. As an application, we provide a conditional ergodic theorem.

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On the quasi-ergodic distribution of absorbing Markov processes

Cited by 16 publications

References 23 publications

Conditioned Lyapunov exponents for random dynamical systems

Conditioned Lyapunov exponents for random dynamical systems

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

Uniform convergence to the $Q$-process

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