Functional limit theorem for occupation time processes of intermittent maps

Sera, Toru

doi:10.1088/1361-6544/ab5ceb

Cited by 11 publications

(6 citation statements)

References 52 publications

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“…of Thaler's arcsine and Aaronson's Darling-Kac laws. Note that the paper [7] contains a good summary of earlier results.…”

Section: Main Results: Hata Mapmentioning

confidence: 95%

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Arcsine and Darling--Kac laws for piecewise linear random interval maps

Hata¹,

Yano²

2021

Preprint

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“…of Thaler's arcsine and Aaronson's Darling-Kac laws. Note that the paper [7] contains a good summary of earlier results.…”

Section: Main Results: Hata Mapmentioning

confidence: 95%

“…The arcsine and Darling-Kac laws for dynamical systems have been developed by many authors. Recently Sera [7] has obtained a functional and joint-distributional generalization…”

Section: Acknowledgementmentioning

confidence: 99%

Arcsine and Darling--Kac laws for piecewise linear random interval maps

Hata¹,

Yano²

2021

Preprint

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“…This was recently generalized in [16] to multi-ray settings by Sera and the fourth author, by using ideas from excursion theory. See also [2,3,15,18] and reference therein.…”

Section: Introductionmentioning

confidence: 99%

Arcsine law for random dynamics with a core

et al. 2023

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In their recent paper, Hata and Yano (2021 Stoch. Dyn. 2350006) first gave an example of random iterations of two piecewise linear interval maps without (deterministic) indifferent periodic points for which the arcsine law—a characterization of intermittent dynamics in infinite ergodic theory—holds. The key in the proof of the result is the existence of a Markov partition preserved by each interval maps. In the present paper, we give a class of random iterations of two interval maps without indifferent periodic points but satisfying the arcsine law, by introducing a concept of core random dynamics. As applications, we show that the generalized arcsine law holds for generalized Hata–Yano maps and piecewise linear versions of Gharaei–Homburg maps, both of which do not have a Markov partition in general.

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“…The ergodic properties of dynamical systems with infinite invariant densities have been established in infinite ergodic theory [12][13][14][15][16][17][18], where distributional limit theorems for time-averaged quantities play an important role. The distributional limit theorems state that time-averaged observables obtained with single trajectories show trajectory-to-trajectory fluctuations.…”

Section: Introductionmentioning

confidence: 99%

Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling

Akimoto,

Barkai,

Radons

2022

Preprint

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We compare ergodic properties of the kinetic energy for three stochastic models of subrecoil-lasercooled gases. One model is based on a heterogeneous random walk (HRW), another is an HRW with long-range jumps (the exponential model), and the other is a mean-field-like approximation of the exponential model (the deterministic model). All the models show an accumulation of the momentum at zero in the long-time limit, and a formal steady state cannot be normalized, i.e., there exists an infinite invariant density. We obtain the exact form of the infinite invariant density and the scaling function for the exponential and deterministic models and devise a useful approximation for the momentum distribution in the HRW model. While the models are kinetically non-identical, it is natural to wonder whether their ergodic properties share common traits, given that they are all described by an infinite invariant density. We show that the answer to this question depends on the type of observable under study. If the observable is integrable, the ergodic properties such as the statistical behavior of the time averages are universal as they are described by the Darling-Kac theorem. In contrast, for non-integrable observables, the models in general exhibit non-identical statistical laws. This implies that focusing on non-integrable observables, we discover non-universal features of the cooling process, that hopefully can lead to a better understanding of the particular model most suitable for a statistical description of the process. This result is expected to hold true for many other systems, beyond laser cooling.

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Functional limit theorem for occupation time processes of intermittent maps

Cited by 11 publications

References 52 publications

Arcsine and Darling--Kac laws for piecewise linear random interval maps

Arcsine and Darling--Kac laws for piecewise linear random interval maps

Arcsine law for random dynamics with a core

Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling

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