Pith. sign in

REVIEW 3 major objections 5 minor 47 references

Jumping for diffusion in random metastable systems

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Random metastable systems converge to one averaged Markov jump process, and their quenched diffusion coefficient follows in closed form.

desk verdict Solid random generalization of Dolgopyat–Wright with a nice closed-form formula, but two missing hypotheses (uniform perturbation theorem and mixing) need to be supplied before the main theorems are airtight. read the letter →

arxiv 2505.22996 v2 pith:TKH7JBAW submitted 2025-05-29 math.DS

Cecilia González-Tokman , Joshua Peters This is my paper
classification math.DS MSC 37H0537C3037E0560J2760F05
keywords randommetastabilitydynamicalsystemsPerron-FrobeniusoperatorcocyclesMarkovjumpprocessquenchedcentrallimittheoremdiffusioncoefficientescaperatespairedtentmaps
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper seeks to show that the slow modes of a randomly driven, weakly perturbed expanding interval map — the rare switches between its $m$ metastable regions — are governed, in the small-perturbation limit, by a single averaged continuous-time Markov jump process. The central claim holds for almost every realisation of the driving noise: the joint distribution of rescaled holding times and jump destinations of the random maps converges to that of the averaged process, whose generator is built from the per-state, per-environment escape coefficients $\beta_{i,j,\omega}$. The paper further claims that the variance in the quenched central limit theorem for random observables is given, at leading order in the perturbation, by a resolvent integral of that same averaged generator, so fluctuation sizes can be predicted from averaged escape rates alone. An explicit closed form is computed for the class of random paired tent maps. If correct, the result extends the deterministic metastability theory of interval maps to randomly forced systems and makes the Markov reduction quantitatively predictive.

What carries the argument

The load-bearing objects are the open Perron–Frobenius (density-transfer) operators $L^\varepsilon_{j,\omega}$ acting on functions supported on a single metastable interval $I_j$, with the random hole $H^\varepsilon_{j,\omega}=H^\varepsilon_{j,j-1,\omega}\cup H^\varepsilon_{j,j+1,\omega}$ removed. Their leading Lyapunov multipliers obey the first-order expansion $\lambda^\varepsilon_{j,\omega}=1-\varepsilon(\beta_{j,j-1,\omega}+\beta_{j,j+1,\omega})+o(\varepsilon)$, which turns each environment's escape probability into an exponential holding-time rate; averaging these rates against the noise gives the generator $\bar{G}$ of the limiting Markov jump process. A sequential perturbation theorem for quenched random open dynamical systems supplies that expansion, and a stability theorem for hyperbolic Oseledets splittings (the Lyapunov subspace decomposition of the transfer-operator cocycle) supplies the uniformity over $\omega$ in the spectral data. The comparison of jump laws proceeds by induction on the number of jumps, with a random growth lemma separating essential from inessential visits to the holes, and the ergodic theorem turns fiberwise sums along the orbit of $\omega$ into integrals against $P$.

What would settle it

Compute the quenched escape multiplier numerically for a concrete two-state random family — for instance paired tent maps driven by a Bernoulli shift on two symbols — at several small $\varepsilon$: if $\lambda^\varepsilon_{j,\omega}=1-\varepsilon(\beta_{j,j-1,\omega}+\beta_{j,j+1,\omega})+o(\varepsilon)$ fails for some environment of positive probability, or the error does not vanish uniformly in $\omega$, then the exponential holding-time limit of Theorem 1.2 and the variance formula (3) cannot both hold. A cheaper check is to simulate the first transition of the random maps and test whether $\varepsilon T^\varepsilon_{1,\omega}$ converges, for almost every realisation, to an exponential law of rate $\int_\Omega(\beta_{j,j-1,\omega}+\beta_{j,j+1,\omega})\,dP(\omega)$.

Watch

Extended reading notes

Core claim

The paper's central claim is that the jump process of a random metastable system is asymptotically indistinguishable from an averaged Markov jump process — a continuous-time Markov chain whose holding times are exponential — realisation by realisation of the noise. For systems satisfying (I1)–(I6), (P1)–(P7) and (O1), Theorem 1.2 states that for $P$-almost every environment $\omega$ and any fixed number $p$ of transitions, the $\mu_j$-measure of points whose first $p$ rescaled holding times $\varepsilon T^\varepsilon_{k,\omega}$ fall in prescribed intervals and whose post-jump states are prescribed converges to the corresponding probability under the averaged Markov jump process with generator $\bar{G}_{ij}=\int_\Omega \beta_{i,j,\omega}\,dP(\omega)$ for $i\neq j$. Theorem 1.3 adds that the quenched (realisation-by-realisation) diffusion coefficient of the central limit theorem satisfies $$\lim_{\varepsilon\to 0}\varepsilon(\Sigma^\varepsilon(\tilde\psi^\varepsilon))^2=2\left\langle p\odot\int_\$\Omega$ \Psi_\omega\,dP(\omega),\int_0^\infty $e^{{t\bar{G}}$}\,dt\int_\$\Omega$ \Psi_\omega\,dP(\omega)\right\rangle,$$ where $p$ is the invariant distribution of the averaged chain, $\Psi_\omega(j)=\int_{I_j}\psi_\omega\,\phi_j\,d\mathrm{Leb}$, and $\odot$ is the entrywise (Hadamard) product. For random paired tent maps with random leakage parameters $\varepsilon a_\omega$ and $\varepsilon b_\omega$, this reduces to the explicit value $2\bar a\bar b(\bar\psi_L-\bar\psi_R)^2/(\bar a+\bar b)^3$. The proof reaches these limits through spectral analysis of the open transfer operators of the system, converting per-environment escape rates into the exponential holding times of the averaged process.

Load-bearing premise

The argument rests on the claim that, uniformly across all environments, the one-step probability of escaping a metastable state equals $\varepsilon$ times a fixed rate plus an error that vanishes with $\varepsilon$; this uniform first-order expansion is imported from a perturbation theorem whose uniform version is justified only by 'close inspection of the proof' and also presumes the non-degeneracy condition (O1) and the controlled error in (P4), and if the uniformity fails, the exponential holding times, the jump-process limit, and the diffusion-coefficient formula would all collapse.

Editorial extensions

If this is right

  • The joint law of successive transitions — holding times together with jump destinations — is determined, in the small-perturbation limit, by the averaged escape coefficients $\int_\Omega \beta_{i,j,\omega}\,dP(\omega)$ alone, so the full random map family need not be simulated to predict transition statistics.
  • Formula (3) makes the quenched central-limit-theorem variance a closed-form expression in the averaged generator's resolvent, so for any system in the class the diffusion coefficient is computable from the mean escape coefficients and the mean observable values.
  • For random paired tent maps the variance is exactly $2\bar a\bar b(\bar\psi_L-\bar\psi_R)^2/(\bar a+\bar b)^3$, showing that only the mean leakage parameters and the mean observable values enter, not finer statistics of the environment.
  • For each fixed small $\varepsilon$, the quenched central limit theorem holds for every regular, fibrewise centered observable, so the Markov approximation comes with a bona fide distributional limit at finite perturbation size.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The reduction the paper establishes for jump statistics suggests the same averaged generator should control rare-event asymptotics: the large-deviation rate function of additive observables of the random maps should coincide with that of the averaged Markov jump process — a testable claim the paper does not address.
  • The paper states that the nearest-neighbour restriction on jumps can be relaxed without difficulty; extending formula (3) to general transition graphs would give the analogue of the paired-tent expression for all-to-all coupling, a direct continuation the authors leave implicit.
  • Formula (3) has a practical reading: measuring per-environment escape rates from time-series data and averaging them predicts the diffusion coefficient that a direct simulation of the CLT variance would produce, offering a validation route for numerically generated metastable systems.
  • The quenched, time-dependent centering in the CLT suggests that in applications such as transport in randomly forced ocean gyres, the effective diffusivity is controlled by the averaged Markov generator regardless of how the forcing varies in time.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies random perturbations of a piecewise expanding interval map with m disjoint invariant subintervals, modeling random metastability. The perturbations create random holes that allow jumps between subintervals, and the authors analyze the resulting Perron-Frobenius operator cocycle. Their main results are: Theorem 1.1, a spectral gap and first-order escape-rate expansion for the open random systems on each interval; Theorem 1.2, convergence of the joint distribution of holding times and jump destinations to an averaged continuous-time Markov jump process; and Theorem 1.3, a quenched CLT for fixed hole size together with a small-hole limit formula for the diffusion coefficient in terms of the averaged generator. The paper closes with an application to Horan's random paired tent maps, where the diffusion coefficient is computed in closed form. The proof strategy adapts the deterministic metastability framework of Dolgopyat and Wright, using sequential perturbation theory, Birkhoff averaging, and the random-perturbation results of Crimmins and others.

Significance. If the main theorems hold as stated, the paper would be a substantial extension of the deterministic metastable diffusion-coefficient theory to the quenched random setting, and it would provide a falsifiable, parameter-free prediction for random paired tent maps. The paper is largely self-contained and carefully tracks the dependencies between the random perturbation and the averaged Markov chain; the paired-tent calculation is explicit and the verification of the abstract hypotheses there is a useful contribution. However, the announced results are conditional on several load-bearing gaps identified below: an unstated mixing hypothesis in the tail estimate, an unproved uniform-in-omega perturbation expansion, and an unjustified replacement of occupation probabilities by Markov jump transition probabilities. These gaps affect Theorems 1.2 and 1.3 directly, so the significance is real but currently conditional.

major comments (3)
  1. [§6, Lemma 6.4 and Eq. (55)] The proof of (55) asserts that T^0_j : I_j → I_j is mixing and then concludes exponential decay of L^{0(n)}_j, but mixing is not among the standing assumptions (I1)–(I6). A piecewise expanding map on an invariant interval can have a unique ergodic ACIM and yet fail to be mixing, for example a period-2 exchange of two subintervals with full tent branches; for such a map the decay in (55) is false. Since (55) is the only mechanism used to obtain the contraction in Lemma 6.5 and hence the tail bound in Step 3 of Theorem 1.3, Theorem 1.3 is not established under the assumptions as written. Please add an explicit mixing or aperiodicity hypothesis on each T^0_j, or prove it from the stated assumptions, and verify it for the paired-tent example.
  2. [§4, Remark 4.12 and Lemma 4.16] The key first-order expansion λ^ε_{j,ω} = 1 - ε(β_{j,j-1,ω} + β_{j,j+1,ω}) + o(ε) is obtained by invoking a uniform-in-ω analogue of [5, Theorem 2.1.2], justified only by 'close inspection of the proof.' This uniformity is load-bearing: it feeds into Corollary 4.17, Lemma 5.2, and therefore into the proofs of Theorems 1.2 and 1.3. Please provide a complete proof of the uniform analogue, with the exact conditions under which the error terms are independent of ω, or alternatively verify the hypotheses of [5, Theorem 2.1.2] directly for the random cocycle.
  3. [§6, Step 7 and Eq. (68)] The passage from µ_j(T^{ε(n-2n_0)}_{σ^{n_0}ω}(x) ∈ I_k) to p_{jk}(uδ) in Eq. (68) invokes Theorem 1.2, but Theorem 1.2 only gives convergence of the joint distribution of holding times and jump destinations. It does not directly give convergence of occupation probabilities at intermediate times. The replacement of the occupation probability by (e^{uδ \bar G})_{jk} requires a separate argument, for example by summing over all possible jump histories or by establishing a stronger path-space convergence of the jump processes. Without such an argument, Eq. (68) and hence the diffusion-coefficient formula (3) are not justified.
minor comments (5)
  1. [§5.1] The definition of \bar G appears typeset as \bar G := \bar M^ε - I ε; it should be (\bar M^ε - I)/ε or an equivalent expression.
  2. [§1.1 and Theorem 1.3] Theorem 1.3 begins 'Fix ε > 0' but then contains a limit ε → 0; please clarify that the quenched CLT holds for each sufficiently small fixed ε and that the diffusion-coefficient limit is taken separately.
  3. [§5.2, Lemma 5.2] The sums over n = 0 to t/ε - 1 implicitly assume that t/ε is an integer; adding floor notation would make the statement valid for arbitrary ε.
  4. [§4, Lemma 4.15] The notation \sum_{i=1}^{P} is used without defining P or the intervals S_i^{ε}_{j,k,ω}; please define these before the estimate.
  5. [§7, Eq. (72)] Please check the typesetting of the denominator in the displayed formula; the derivation gives 2 \bar a \bar b/(\bar a+\bar b)^3 (\bar ψ_L - \bar ψ_R)^2, so the displayed denominator should reflect the cube.

Circularity Check

1 steps flagged · score 2.0 of 10

No by-construction circularity; the diffusion and jump formulas are genuine closed-form consequences of the stated model inputs, but two load-bearing steps rest on same-group preprints rather than on independently verified theorems.

  1. self citation load bearing [Section 4, Remark 4.12 and Lemma 4.16; used in Corollary 4.17 and Theorems 1.2 and 1.3.]
    "This result relies on the sequential perturbation theorem [5, Theorem 2.1.2]. Remark 4.12. We note that the conditions of [5, Theorem 2.1.2] are required to hold for each ω ∈ Ω. In our setting, we find that such conditions hold uniformly over ω ∈ Ω away from a P-null set. Upon close inspection of the proof of [5, Theorem 2.1.2], a uniform over ω ∈ Ω away from a P-null set analogue of [5, Theorem 2.1.2] holds."

    The first-order multiplier expansion λε_j,ω = 1 − ε(βj,j−1,ω + βj,j+1,ω) + o(ε), which is the input that makes holding times asymptotically exponential in Theorem 1.2 and determines the generator Ḡ in Theorem 1.3, is established by verifying the hypotheses of [5, Theorem 2.1.2] and importing its conclusion. [5] is a same-group preprint (Atnip, Froyland, Gonzalez-Tokman, Vaienti), and the uniform-in-ω version actually needed is not a stated result there; Remark 4.12 supplies it only by asserting that a careful reading of the proof yields it. Thus the central escape-rate input to both main theorems rests on the authors' own prior work plus an unproved uniformity claim, rather than on an independently verified theorem.

full rationale

The paper's central derivation is not circular by construction. The coefficients βi,j,ω are model inputs specified in (P4) as the first-order coefficients of the hole measures µi(Hε_i,j,ω); they are not fitted constants. Theorems 1.1 through 1.3 form an internal derivation chain: spectral perturbation produces the escape rates, Theorem 1.2 converts those rates into a Markov jump process limit, and Theorem 1.3 computes the diffusion coefficient from that jump process. The paired-tent formula (72) is a parameter-free closed form in terms of ā, b̄, and the observable averages, so it is externally falsifiable rather than equivalent to an input. The main circularity burden is the load-bearing reliance on the same-group preprint [5, Theorem 2.1.2] for the escape-rate expansion, with a uniform analogue asserted only by 'close inspection' in Remark 4.12; the paper likewise uses [30, Theorem 7.2], [30, Lemma 5.2], and [30, Section 8] for the limiting invariant density and paired-tent generator, all by the same author pair and not machine-checked. These are self-citations with substantial independent content, so they do not make the derivations equivalent to their inputs; they are a provenance and verification concern. The missing mixing/aperiodicity hypothesis in Lemma 6.4 is a correctness risk, not a circularity, and is therefore not counted in the score. Overall, no step reduces a central prediction to an input by definition, so the circularity score is low.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new physical entities; the averaged Markov jump process is a construction built from the input escape rates β, not an ad hoc device. The honest ledger is therefore about the modeling assumptions ((I), (P), (O1)) and the imported theorems from [5], [14] and [30], which carry most of the analytic weight.

free parameters (4)
  • β_{i,j,ω} (hole escape-rate functions)
    Defined in (P4) by µ_i(H^ε_{i,j,ω}) = εβ_{i,j,ω} + o(ε), with β in L∞(P). These are inputs describing the leak sizes, not fitted to any output; every conclusion of Theorems 1.2 and 1.3 is expressed through their P-averages.
  • β* (uniform lower bound on all β_{i,j,ω})
    Imposed in (P4) to ensure uniform covering over ω, needed for the quenched CLT [14] and for the uniform error control in Lemma 4.16 and Corollary 4.17.
  • ε (hole size parameter)
    Regime parameter; all results are ε→0 limits, and the diffusion coefficient per unit ε is the computed quantity, so ε acts as the small parameter of the metastable scaling.
  • Nearest-neighbour jump structure (Remark 3.4)
    Assumption that points in I_j map only to I_{j±1} or stay in I_j, making Ḡ tridiagonal; the paper asserts it can be relaxed without difficulty, but all stated theorems use it.
assumptions (7)
  • domain assumption (I1)-(I6): piecewise C2, uniformly expanding, m invariant subintervals, unique ACIM on each I_j, no critical-set returns to infinitesimal holes, positive densities at holes
    Defines the unperturbed metastable map (Section 3.1); (I5)-(I6) are used in the density and hole estimates of Lemmas 4.13-4.15 and Lemma 5.8.
  • domain assumption (P1)-(P7): ergodic P-preserving base, finite-range P-continuous C2-small perturbations, convergent holes, escape-rate scaling (P4), uniform Lasota-Yorke, unique RACIM, boundary conditions
    Defines the random perturbation class (Section 3.2). Finite range (P1) is used repeatedly to make error constants uniform in ω; (P6) gives the unique ergodic regime.
  • ad hoc to paper (O1) non-vanishing surviving branches: n' with Λ^{-n'} < 1/9 and ess inf min_Z Leb_j(Z) > 0
    Imposed in Section 4 to secure the uniform Lasota-Yorke bound in Lemma 4.7 and the spectral gap; surrogate conditions (6a)-(6b) are offered, but (O1) itself is tailored to the proof machinery.
  • standard math [5, Theorem 2.1.2] and its asserted uniform-in-ω analogue
    Used in Lemma 4.16 to derive λ^ε_{j,ω} = 1 - ε(β_{j,j-1,ω}+β_{j,j+1,ω}) + o(ε), the backbone of the exponential holding-time asymptotics; the uniform analogue is asserted in Remark 4.12 without proof.
  • standard math [30, Lemma 5.2, Theorems 7.2 and 8.1] (same authors' prior averaging results)
    Lemma 5.2 imports the Birkhoff-average computation for the jump sums; Theorem 7.2 supplies the limiting density weights p_j; Theorem 8.1 supplies the paired-tent stationary measure used in Section 7.
  • standard math [14, Theorem B] quenched CLT for random expanding systems
    Provides the fixed-ε normal limit (2); its hypotheses (uniform covering, fibrewise centering, regularity) are verified or imposed in Section 6.
  • domain assumption Exponential mixing of each T^0|I_j (rate bound (55))
    Lemma 6.4 needs exponential contraction of L^{0(n)}_j, which is stronger than the unique-ACIM assumption (I4); this hypothesis is not listed in (I1)-(I6).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Jumping for diffusion in random metastable systems." pith.science (2026). https://pith.science/paper/TKH7JBAW

@misc{pith2026250522996,
  author       = {Pith},
  title        = {Pith review of: Jumping for diffusion in random metastable systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TKH7JBAW}},
  note         = {Machine review of arXiv:2505.22996}
}
read the original abstract

Random metastability occurs when an externally forced or noisy system possesses more than one state of apparent equilibrium. This work investigates fluctuations in a class of random dynamical systems, arising from randomly perturbing a piecewise smooth expanding interval map with more than one invariant subinterval. Upon perturbation, this invariance is destroyed, allowing trajectories to switch between subintervals, giving rise to metastable behaviour. We show that the distributions of jumps of a time-homogeneous Markov chain approximate the distributions of jumps for random metastable systems. Additionally, we demonstrate that this approximation extends to the diffusion coefficient for (random) observables of such systems. As an example, our results are applied to Horan's random paired tent maps.

Figures

Figures reproduced from arXiv: 2505.22996 by the authors.

Figure 1
Figure 1. −1 1 −1 1 x T0,0 −1 1 −1 1 b −a x Ta,b [PITH_FULL_IMAGE:figures/full_fig_p056_1.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

47 extracted references · 46 canonical work pages

  1. [13]

    Dolgopyat and P

    D. Dolgopyat and P. Wright. The diffusion coefficient for piecewise expanding maps of the interval with metastable states. Stoch. Dyn., 12(1):1150005, 13, 2012

    work page 2012

  2. [1]

    Abdelkader and R

    M. Abdelkader and R. Aimino. On the quenched central limit theorem for random dynamical systems. J. Phys. A , 49(24):244002, 13, 2016

    work page 2016

  3. [2]

    Aimino, M

    R. Aimino, M. Nicol, and S. Vaienti. Annealed and quenched limit theorems for random expanding dynamical systems. Probab. Theory Related Fields, 162(1-2):233–274, 2015

    work page 2015

  4. [3]

    L. Arnold. Random dynamical systems. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998

    work page 1998

  5. [4]

    Atnip, G

    J. Atnip, G. Froyland, C. Gonz´ alez-Tokman, and S. Vaienti. Thermodynamic formalism for random weighted covering systems. Comm. Math. Phys. , 386(2):819–902, 2021

    work page 2021

  6. [5]

    Atnip, G

    J. Atnip, G. Froyland, C. Gonzalez-Tokman, and S. Vaienti. Thermodynamic formalism and pertur- bation formulae for quenched random open dynamical systems, 2023. arXiv: 2307.00774, to appear in Dissertationes Mathematicae

    arXiv 2023

  7. [6]

    Bahsoun and S

    W. Bahsoun and S. Vaienti. Metastability of certain intermittent maps. Nonlinearity, 25(1):107–124, 2012

    work page 2012

  8. [7]

    Bahsoun and S

    W. Bahsoun and S. Vaienti. Escape rates formulae and metastability for randomly perturbed maps. Nonlinearity, 26(5):1415–1438, 2013

    work page 2013

Show all 47 references
  1. [8]

    V. Baladi. Positive transfer operators and decay of correlations , volume 16 of Advanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River Edge, NJ, 2000

    2000

  2. [9]

    Butterley and P

    O. Butterley and P. Eslami. Exponential mixing for skew products with discontinuities. Trans. Amer. Math. Soc., 369(2):783–803, 2017

    2017

  3. [10]

    J. Buzzi. Exponential decay of correlations for random Lasota-Yorke maps. Comm. Math. Phys. , 208(1):25–54, 1999

    1999

  4. [11]

    Crimmins

    H. Crimmins. Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles. Discrete Contin. Dyn. Syst. , 42(6):2795–2857, 2022

    2022

  5. [12]

    Dellnitz, G

    M. Dellnitz, G. Froyland, C. Horenkamp, K. Padberg-Gehle, and A. S. Gupta. Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators. Nonlinear Processes in Geophysics, 16(6):655–663, 2009

    2009

  6. [14]

    Dragiˇ cevi´ c, G

    D. Dragiˇ cevi´ c, G. Froyland, C. Gonz´ alez-Tokman, and S. Vaienti. A spectral approach for quenched limit theorems for random expanding dynamical systems. Comm. Math. Phys., 360(3):1121–1187, 2018

    2018

  7. [15]

    Dragiˇ cevi´ c and J

    D. Dragiˇ cevi´ c and J. Sedro. Quenched limit theorems for expanding on average cocycles.Stoch. Dyn., 23(6):Paper No. 2350046, 42, 2023

    2023

  8. [16]

    Eslami and P

    P. Eslami and P. G´ ora. Stronger Lasota-Yorke inequality for one-dimensional piecewise expanding transformations. Proc. Amer. Math. Soc. , 141(12):4249–4260, 2013

    2013

  9. [17]

    Ferguson and M

    A. Ferguson and M. Pollicott. Escape rates for Gibbs measures. Ergodic Theory Dynam. Systems , 32(3):961–988, 2012

    2012

  10. [18]

    Froyland, C

    G. Froyland, C. Gonz´ alez-Tokman, and A. Quas. Stability and approximation of random invariant densities for Lasota-Yorke map cocycles. Nonlinearity, 27(4):647–660, 2014

    2014

  11. [19]

    Froyland, S

    G. Froyland, S. Lloyd, and A. Quas. Coherent structures and isolated spectrum for Perron-Frobenius cocycles. Ergodic Theory Dynam. Systems , 30(3):729–756, 2010

    2010

  12. [20]

    Froyland, S

    G. Froyland, S. Lloyd, and A. Quas. A semi-invertible Oseledets theorem with applications to transfer operator cocycles. Discrete Contin. Dyn. Syst. , 33(9):3835–3860, 2013

    2013

  13. [21]

    Froyland, K

    G. Froyland, K. Padberg, M. H. England, and A. M. Treguier. Detection of coherent oceanic structures via transfer operators. Phys. Rev. Lett. , 98:224503, 2007. Jumping for diffusion in random metastable systems 60

    2007

  14. [22]

    Froyland, N

    G. Froyland, N. Santitissadeekorn, and A. Monahan. Optimally coherent sets in geophysical flows: A transfer-operator approach to delimiting the stratospheric polar vortex. Phys. Rev. E, 82:056311, 2010

    2010

  15. [23]

    Froyland, N

    G. Froyland, N. Santitissadeekorn, and A. Monahan. Transport in time-dependent dynamical systems: finite-time coherent sets. Chaos, 20(4):043116, 10, 2010

    2010

  16. [24]

    Froyland and O

    G. Froyland and O. Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynam- ical systems. Discrete and Continuous Dynamical Systems - B , 14(2):457–472, 2010

    2010

  17. [25]

    Froyland and O

    G. Froyland and O. Stancevic. Metastability, Lyapunov exponents, escape rates, and topological entropy in random dynamical systems. Stoch. Dyn., 13(4):1350004, 26, 2013

    2013

  18. [26]

    Froyland, R

    G. Froyland, R. M. Stuart, and E. van Sebille. How well-connected is the surface of the global ocean? Chaos, 24(3):033126, 10, 2014

    2014

  19. [27]

    Gonz´ alez-Tokman, B

    C. Gonz´ alez-Tokman, B. R. Hunt, and P. Wright. Approximating invariant densities of metastable systems. Ergodic Theory Dynam. Systems , 31(5):1345–1361, 2011

    2011

  20. [28]

    Gonz´ alez-Tokman and A

    C. Gonz´ alez-Tokman and A. Quas. Lyapunov exponents for transfer operator cocycles of metastable maps: a quarantine approach. Trans. Moscow Math. Soc., 82:65–76, 2021

    2021

  21. [29]

    Gonz´ alez-Tokman and A

    C. Gonz´ alez-Tokman and A. Quas. Stability and collapse of the Lyapunov spectrum for Perron- Frobenius operator cocycles. J. Eur. Math. Soc. (JEMS) , 23(10):3419–3457, 2021

    2021

  22. [30]

    Gonz´ alez-Tokman and J

    C. Gonz´ alez-Tokman and J. Peters. Averaging for random metastable systems. arXiv, 2024. arXiv: 2406.09672

    2024

  23. [31]

    J. Horan. Dynamical spectrum via determinant-free linear algebra, 2020. arXiv: 2001.06788

    2020 arXiv

  24. [32]

    J. Horan. Asymptotics for the second-largest Lyapunov exponent for some Perron-Frobenius operator cocycles. Nonlinearity, 34(4):2563–2610, 2021

    2021

  25. [33]

    G. Keller. Stochastic stability in some chaotic dynamical systems. Monatsh. Math. , 94(4):313–333, 1982

    1982

  26. [34]

    G. Keller. Rare events, exponential hitting times and extremal indices via spectral perturbation. Dyn. Syst., 27(1):11–27, 2012

    2012

  27. [35]

    Keller and C

    G. Keller and C. Liverani. Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. , 135(3):519–534, 2009

    2009

  28. [36]

    J. F. C. Kingman. Subadditive ergodic theory. Ann. Probability, 1:883–909, 1973

    1973

  29. [37]

    Lasota and J

    A. Lasota and J. A. Yorke. On the existence of invariant measures for piecewise monotonic transfor- mations. Trans. Amer. Math. Soc. , 186:481–488 (1974), 1973

    1974

  30. [38]

    Liverani

    C. Liverani. Decay of correlations for piecewise expanding maps. J. Statist. Phys. , 78(3-4):1111–1129, 1995

    1995

  31. [39]

    Liverani

    C. Liverani. Central limit theorem for deterministic systems. In International Conference on Dynamical Systems (Montevideo, 1995) , volume 362 of Pitman Res. Notes Math. Ser. , pages 56–75. Longman, Harlow, 1996

    1995

  32. [40]

    J. R. Norris. Markov chains, volume 2 of Cambridge Series in Statistical and Probabilistic Mathematics . Cambridge University Press, Cambridge, 1998. Reprint of 1997 original

    1998

  33. [41]

    T. Ohno. Asymptotic behaviors of dynamical systems with random parameters. Publ. Res. Inst. Math. Sci., 19(1):83–98, 1983

    1983

  34. [42]

    V. I. Oseledec. A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems. Trudy Moskov. Mat. Obˇ sˇ c., 19:179–210, 1968

    1968

  35. [43]

    Sch¨ utte, W

    C. Sch¨ utte, W. Huisinga, and P. Deuflhard. Transfer operator approach to conformational dynamics in biomolecular systems. In Ergodic theory, analysis, and efficient simulation of dynamical systems , pages 191–223. Springer, Berlin, 2001

    2001

  36. [44]

    Sch¨ utte and M

    C. Sch¨ utte and M. Sarich. Metastability and Markov state models in molecular dynamics , volume 24 of Courant Lecture Notes in Mathematics . Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2013. Modeling, analysis, algorith...

    2013

  37. [45]

    Thieullen

    P. Thieullen. Fibr´ es dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 4(1):49–97, 1987

    1987

  38. [46]

    Viana and K

    M. Viana and K. Oliveira. Foundations of ergodic theory, volume 151 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016

    2016

  39. [47]

    G. G. Yin and Q. Zhang. Continuous-time Markov chains and applications , volume 37 of Stochas- tic Modelling and Applied Probability . Springer, New York, second edition, 2013. A two-time-scale approach

    2013

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.