Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories

Benaı̈m, Michel; Bouguet, Florian; Cloez, Bertrand

doi:10.1214/17-aap1275

Cited by 7 publications

(22 citation statements)

References 35 publications

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“…In this section, we prove Theorem 2.9 using results from [BBC16], based on the theory of asymptotic pseudotrajectories for inhomogeneous-time Markov chains. Indeed, with the convention and define the piecewise-constant processes…”

Section: Asymptotic Pseudotrajectories In the Non-standard Settingmentioning

confidence: 99%

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Fluctuations of the empirical measure of freezing Markov chains

Bouguet¹,

Cloez²

2018

Electron. J. Probab.

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In this work, we consider a finite-state inhomogeneous-time Markov chain whose probabilities of transition from one state to another tend to decrease over time. This can be seen as a cooling of the dynamics of an underlying Markov chain. We are interested in the long time behavior of the empirical measure of this freezing Markov chain. Some recent papers provide almost sure convergence and convergence in distribution in the case of the freezing speed 1/n θ , with different limits depending on θ < 1, θ = 1 or θ > 1. Using stochastic approximation techniques, we generalize these results for any freezing speed, and we obtain a better characterization of the limit distribution as well as rates of convergence as well as functional convergence.

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Section: Asymptotic Pseudotrajectories In the Non-standard Settingmentioning

confidence: 99%

“…for any sequence of positive weights (ω n ) n≥1 , as in [BC15, Remark 1.1] or [BBC16, Section 3.1]. Then, we define γ n = n k=1 ω k , and Theorem 2.11 below still holds with the bound BBC16], and given sequences (γ n ) n≥1 , ( n ) n≥1 , we define the following parameter which rules the speed of convergence in the context of standard fluctuations:…”

mentioning

confidence: 99%

Fluctuations of the empirical measure of freezing Markov chains

Bouguet¹,

Cloez²

2018

Electron. J. Probab.

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“…is known, we are done, and this latter type of limit of "pseudo-trajectories" is what has been derived in [1] under some suitable assumptions. In summary, [2] provides a very valuable complement to [7].…”

Section: 2mentioning

confidence: 96%

The coin-turning walk and its scaling limit

Engländer¹,

Volkov²,

Wang³

2020

Electron. J. Probab.

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Let S be the random walk obtained from "coin turning" with some sequence {p n } n≥2 , as introduced in [7]. In this paper we investigate the scaling limits of S in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics.We prove that an invariance principle, albeit with a non-classical scaling, holds for "not too small" sequences, the order const•n −1 (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold.In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed.We also investigate the recurrence of the walk and its scaling limit, as well as the ergodicity and mixing of the nth step of the walk.

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“…This idea goes back to [4]. In the case of systems driven by a Lévy process (with γ fixed), [18] gives a precise estimate of the error and compares the approximation obtained by truncation as in equation (6) with the one obtained by adding a Gaussian noise as in equation (7). An enlightening discussion on the complexity of the two methods is also provided.…”

Section: Introductionmentioning

confidence: 99%

“…L n ) is the semigroup (resp. infinitesimal generator) associated with (7) and P (resp. L) is the semigroup (resp.…”

Section: Introductionmentioning

confidence: 99%

Regularity and stability for the semigroup of jump diffusions with state-dependent intensity

Bally¹,

Goreac²,

Rabiet³

2018

Ann. Appl. Probab.

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We consider stochastic differential systems driven by a Brownian motion and a Poisson point measure where the intensity measure of jumps depends on the solution. This behavior is natural for several physical models (such as Boltzmann equation, piecewise deterministic Markov processes, etc). First, we give sufficient conditions guaranteeing that the semigroup associated with such an equation preserves regularity by mapping the space of the of k-times differentiable bounded functions into itself. Furthermore, we give an explicit estimate of the operator norm. This is the key-ingredient in a quantitative Trotter-Kato-type stability result: it allows us to give an explicit estimate of the distance between two semigroups associated with different sets of coefficients in terms of the difference between the corresponding infinitesimal operators. As an application, we present a method allowing to replace "small jumps" by a Brownian motion or by a drift component. The example of the 2D Boltzmann equation is also treated in all detail.

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Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories

Cited by 7 publications

References 35 publications

Fluctuations of the empirical measure of freezing Markov chains

Fluctuations of the empirical measure of freezing Markov chains

The coin-turning walk and its scaling limit

Regularity and stability for the semigroup of jump diffusions with state-dependent intensity

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