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REVIEW 2 major objections 7 minor 43 references

Impact of spinning on the early-warning signs in non-Markovian stochastic systems

T0 review · 2 major / 7 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Spinning erases memory from early-warning signals

desk verdict Solid contribution: new scaling laws for non-Markovian EWSs across full Hurst range, with the Hopf H-independence result being the standout. Deserves a serious referee. read the letter →

arxiv 2607.06428 v1 pith:NYEILOFZ submitted 2026-07-07 math.PR math.DS

Paolo Bernuzzi , Alexandra Blessing , Dennis Rudik This is my paper · ORCID
classification math.PRmath.DS MSC 60G2260H1037H20
keywords early-warningsignalsHopfbifurcationfractionalBrownianmotionnon-Markoviannoiseautocovariancespectraldensityfast-slowsystemscriticaltransitions
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 proves that when a system approaches a Hopf bifurcation, the rotational motion inherent to that bifurcation acts as a mixing mechanism that erases the memory of the driving noise. For one-dimensional bifurcations (fold, transcritical, pitchfork), the scaling laws of early-warning observables like autocovariance and spectral density depend explicitly on the Hurst index H, which parametrizes how much memory the noise carries. The authors show that near a Hopf bifurcation this dependence vanishes: the autocovariance diverges as |A(λ)|^{-1} and the spectral density at the rotational frequencies diverges as |A(λ)|^{-2}, both independent of H. The paper establishes these results for fractional Brownian motion across the full range H in (0,1), extending prior work that was restricted to H > 1/2, and further extends the analysis to red noise and fractional Ornstein-Uhlenbeck forcings. The central mechanism is that the imaginary part B(λ) of the eigenvalue, which governs rotation, remains nonzero at the bifurcation threshold, causing the memory term P in the autocovariance decomposition to converge rather than diverge, leaving only the mitigation term (loss of stability) to control the scaling.

What carries the argument

The autocovariance decomposition into three factors, the memory function P, and the spectral density formula

What would settle it

If the memory term P failed to converge at the Hopf threshold for some H in (0,1), the H-independence of the scaling laws would break down.

Watch

Extended reading notes

Core claim

The decomposition of the time-asymptotic autocovariance into three factors (masking, memory, mitigation) reveals that rotational dynamics near a Hopf bifurcation force the memory factor to remain bounded, so the scaling exponent of early-warning signals becomes independent of the Hurst index. This is in sharp contrast to one-dimensional codimension-1 bifurcations where the memory factor diverges and the exponent is governed by H. The spectral density develops peaks at the rotational frequencies ±B(λ) that diverge as |A(λ)|^{-2} regardless of noise coloring, providing an observable that is immune to the masking (color blindness) that can defeat variance-based indicators when the noise is a fO

Load-bearing premise

The entire analysis rests on the linearization of the fast subsystem along the attracting critical manifold, which is valid only in a neighborhood of the stable branch. Since early-warning signals are most needed near the bifurcation threshold where nonlinear effects become important, the derived scaling laws hold only in a regime bounded away from the critical point, and the transition between the linearized and nonlinear regimes is not rigorously characterized.

Editorial extensions

If this is right

  • For systems approaching a Hopf bifurcation, the Hurst index of the driving noise need not be estimated to interpret early-warning signals, simplifying practical deployment.
  • The spectral density evaluated at the rotational frequency ±B(λ) provides a noise-robust early-warning signal that works even for fractional Ornstein-Uhlenbeck noise where variance-based indicators fail due to masking.
  • The transition between the H-dependent scaling regime (far from bifurcation) and the H-independent regime (closer to bifurcation) occurs when the real part of the eigenvalue becomes comparable in magnitude to the imaginary part, giving a concrete criterion for when rotation dominates.
  • The unmasking effect of rotation suggests that Hopf bifurcations in systems with colored noise are intrinsically more detectable than fold-type bifurcations, which has implications for monitoring tipping elements in climate and ecological systems.

Reading between the lines

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

  • The memory-erasure mechanism should generalize to any bifurcation where eigenvalues cross the imaginary axis away from the origin, including certain codimension-2 bifurcations with rotational components.
  • The three-stage regime observed in simulations (H-dependent scaling, then H-independent scaling, then nonlinear damping) suggests that the practical detectability window for Hopf early-warning signals is widest when the noise intensity is small enough that the linearized regime persists close to the bifurcation threshold.
  • If the rotational frequency B(λ*) is very small, the memory-erasure effect weakens and the system may behave more like the one-dimensional case, creating a continuous interpolation between the two scaling regimes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 7 minor

Summary. This paper studies early-warning signals (EWSs) for critical transitions in fast-slow stochastic systems driven by non-Markovian noise (fractional Brownian motion, fractional Ornstein-Uhlenbeck processes, and red noise). The main contribution is a comparison of scaling laws for autocovariance, autocorrelation, and spectral density near codimension-1 bifurcations (fold, transcritical, pitchfork) versus Hopf bifurcations. For codimension-1 bifurcations, the authors extend existing results to the full range H∈(0,1) and show that scaling exponents depend explicitly on the Hurst index H. For Hopf bifurcations, they prove that rotational dynamics induce H-independent scaling laws (|A(λ)|^{-1} for autocovariance, |A(λ)|^{-2} for spectral density at ±B(λ)). The analysis covers fBm, red noise, and fOU forcings, and includes numerical validation on a Stommel-Cessi AMOC model and a Hopf normal form system. The proofs in Appendices A and B use the Mandelbrot-van Ness representation, Itô isometry, and Taylor expansion techniques.

Significance. The paper makes a genuine contribution by systematically comparing how bifurcation type and noise memory interact to shape EWS scaling laws. The H-independence result for Hopf bifurcations (Theorems 4.1 and 4.5) is a non-trivial finding: the rotational dynamics suppress the memory signature of the driving fBm, which has practical implications for EWS interpretation. The extension to H∈(0,1/2) (anti-persistent noise) for codimension-1 bifurcations fills a gap in the literature. The decomposition of the autocovariance into masking, memory, and mitigation terms (Lemma 3.1, equation 3.3) is a useful conceptual framework. The spectral density results for fOU noise (showing that S(δ) remains effective even when variance-based EWSs fail due to color blindness) are practically relevant. The numerical simulations on the AMOC model and Hopf system provide reasonable cross-validation of the analytic predictions, including the three-regime behavior (1D scaling, H-independent scaling, nonlinear damping) visible in Figure 10.

major comments (2)
  1. §4.1, Theorem 4.1, Eqs. (4.2)–(4.3): The H-independent scaling |V_∞(τ)[v1,v2]| ≍ |A(λ)|^{-1} is stated to hold for 'almost every τ≥0' and for v1,v2 satisfying condition (4.3). The numerical simulations in Figure 10(c) show that at τ=0, the mixed-mode autocovariance exhibits no intermediate regime with slope -1 at all, which is consistent with (4.3) failing at that specific τ. However, the set of excluded τ values is characterized only implicitly through (4.3), which depends on the eigenvector structure of M(λ*) and the noise covariance Q. The paper would benefit from a more explicit characterization of when (4.3) fails, or at minimum a sharper statement of the measure of the excluded set. As it stands, a practitioner cannot determine a priori which lag times are admissible for a given system. This is load-bearing for the paper's central claim because the practical utility of the autocovE
  2. §6.1, discussion following (A2): The linearization approximation (2.10) is acknowledged to be valid only in a neighborhood of the stable branch, and the authors note that nonlinear effects eventually halt the divergence of EWSs near the critical threshold. The numerical results (Figure 10) reveal three regimes: (1) 1D scaling |A|^{-2H} when |A(λ)| >> |B(λ)|, (2) H-independent scaling |A|^{-1} when |A(λ)| ~ |B(λ)|, and (3) nonlinear damping. The transition from regime (1) to regime (2) is not rigorously characterized. Since the H-independent scaling is the paper's central novel claim, the absence of any analytic bound on when regime (2) begins (in terms of |A(λ)|/|B(λ)|) weakens the practical applicability of the result. The authors should at least provide a heuristic criterion or an explicit estimate for the crossover.
minor comments (7)
  1. Table 1: The entry for fOU variance reads 'Conv(2-2H)'. For H∈(0,1/2), the exponent 2-2H is in (1,2), so the variance converges to 0 as λ→λ*. This is consistent with the color blindness discussed in Section 5, but a brief footnote in the table caption clarifying that 'Conv' here means 'converges to zero' (as opposed to converges to a nonzero constant) would help the reader.
  2. §3.1, Eq. (3.6): The phrase 'almost every τ≥0' is used without specifying the measure. It should be clarified that this refers to Lebesgue-a.e. τ.
  3. §2.4: The notation ζ_1(λ) = A(λ) + iB(λ) = ζ_2(λ) appears to contain a typo; presumably ζ_2(λ) = A(λ) - iB(λ), consistent with the complex conjugate eigenvalue pair.
  4. §5.2, Table 5 caption: The caption states the table applies to 'equivalently for (5.7)', but the text preceding the table discusses only (5.6). A cross-reference clarifying that the same exponents hold for the fOU system (5.7) in the (S2) setting would improve readability.
  5. §6.2, Fig. 4(c): The text states that for H>1/2, the maximum of the SD 'is not attained' analytically but is computed numerically, yielding a slope of -2. This should be stated more carefully: the numerical maximum is finite due to discretization, but the true SD has a singularity at ω=0 for H>1/2.
  6. Several typos: 'reults' (abstract), 'Ornestein-Uhlenbeck' (Section 5.1), 'explicitate' (§6.1, A2), 'ulterior complexity' (§4.1). Also, 'Gausssian' in the reference to Rosenblatt processes on p.3.
  7. References [5] and [40] appear to be by one of the authors (Bernuzzi) and are cited as PhD thesis and recent preprint respectively. Ensuring these are accessible (e.g., via arXiv or repository) would be helpful.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies the paper's main contributions: the H-independence result for Hopf bifurcations, the extension to anti-persistent noise, and the decomposition into masking/memory/mitigation terms. Both major comments concern the practical applicability of the autocovariance EWS in the Hopf setting. We address each below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: §4.1, Theorem 4.1, Eqs. (4.2)–(4.3): The H-independent scaling holds for 'almost every τ≥0' and v1,v2 satisfying (4.3), but the excluded set is characterized only implicitly. A practitioner cannot determine a priori which lag times are admissible. Request for a more explicit characterization or a sharper statement of the measure of the excluded set.

    Authors: The referee is correct that condition (4.3) characterizes the admissible lag times only implicitly, and that this limits practical applicability. We will revise the manuscript to address this in two ways. First, we will add an explicit sufficient condition: condition (4.3) is guaranteed to hold whenever v1 and v2 both have nonzero projection onto at least one of the eigenvectors e_1(λ*) or e_2(λ*) and the corresponding diagonal masking terms ⟨e*_j(λ*), Q e*_j(λ*)⟩ are nonzero (which holds generically when Q is positive definite). This covers the practically relevant case of observing along coordinate directions when the noise is non-degenerate. Second, we will state more sharply that the excluded set of τ values is discrete (it consists of isolated points where a specific oscillatory combination vanishes), so its Lebesgue measure is zero and any randomly chosen τ is admissible with probability one. We will also note that the numerical observation in Figure 10(c) at τ=0 for the mixed mode is precisely one such isolated exclusion, and that τ=0.5 recovers the H-independent regime, consistent with the theory. We agree that the original presentation was insufficiently explicit for practitioners and will revise accordingly. revision: partial

  2. Referee: §6.1, discussion following (A2): The transition from regime (1) (1D scaling |A|^{-2H} when |A(λ)| >> |B(λ)|) to regime (2) (H-independent scaling |A|^{-1} when |A(λ)| ~ |B(λ)|) is not rigorously characterized. No analytic bound on when regime (2) begins. Request for a heuristic criterion or explicit estimate for the crossover.

    Authors: The referee identifies a genuine gap between the analytic results and their practical applicability. We will add a heuristic crossover criterion. The key observation is that the memory term P(ζ_1(λ), ζ_2(λ), H, τ) in (4.1) converges to a finite, nonzero limit as λ→λ* precisely because B(λ*)≠0, whereas in the one-dimensional case the analogous term diverges. The crossover from regime (1) to regime (2) occurs when the imaginary part B(λ) of the eigenvalue becomes comparable in magnitude to the real part |A(λ)|, because at that point the oscillatory phase e^{iB(λ)τ} in the memory term begins to produce cancellations that suppress the H-dependent growth. Concretely, the heuristic criterion is |A(λ)| ≲ |B(λ)|, or equivalently |A(λ)|/|B(λ)| ≲ 1. We will verify that this is consistent with the numerical results in Figure 10, where ω_0 = B(λ*) = 1 and the crossover occurs near |A(λ)| ≈ 1. We will also note that the width of the intermediate regime depends on the noise intensity σ, as visible in the comparison between panels (a) and (b) of Figure 10. We agree that a fully rigorous bound on the crossover would require controlling the remainder terms in the Taylor expansion underlying Lemma 4.2 uniformly in the ratio |A(λ)|/|B(λ)|, which is beyond the scope of the current analysis. We will state this limitation explicitly. revision: partial

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; scaling laws derived from first principles with minor self-citation for methods.

full rationale

The paper derives scaling laws for early-warning signals (autocovariance, autocorrelation, spectral density) from the covariance structure of fractional Brownian motion, solutions to linearized SDEs, and Fourier analysis. No parameters are fitted to target results: H is an input parameter of the noise model, σ is structural noise intensity, and A(λ), B(λ) are drift eigenvalues. The central H-independence claim for Hopf bifurcations (Theorems 4.1, 4.5) follows from the mathematical fact that B(λ*) ≠ 0 keeps the memory term P(ζ, ζ, H, τ) bounded (eq. B.2), which is a genuine analytical consequence, not a definition or fit. Self-citations [5, 6, 7, 8] are used for methods (Mandelbrot-van-Ness representation, ergodic properties, estimator consistency) but the central scaling results are proven independently in Appendices A and B. The one minor concern is that Lemma 3.1's proof (Proof A.1) extends methods from [29] (Kubilius-Mishura-Ralchenko) and the H=1/2 case from [6], but these are standard tools, not load-bearing claims that would make the present results circular. The numerical simulations (Figures 4-12) validate the analytic predictions against independent models (Stommel-Cessi AMOC, Hopf normal form) without fitting parameters to reproduce the scaling exponents. The derivation chain is self-contained against external benchmarks.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities, particles, forces, or dimensions. All mathematical objects (fBm, fOU, spectral density, incomplete gamma functions) are standard. The 'masking,' 'memory,' and 'mitigation' terms are interpretive labels for factors in the covariance formula (3.3), not new mathematical objects. The free parameters (H, σ, μ, ε) are input parameters of the stochastic model, not fitted constants. The analysis is parameter-free in the sense that scaling exponents are derived, not fitted.

free parameters (5)
  • Hurst index H
    Input parameter of the fBm noise model, not fitted to data. Takes values in (0,1) and is treated as a known property of the noise.
  • Noise intensity σ
    Structural parameter of the SDE, not fitted. Set to specific values in simulations (σ=0.001 for AMOC, σ₁=0.01, σ₂=0.05 for Hopf).
  • fOU drift parameter μ
    Parameter of the fractional Ornstein-Uhlenbeck process (5.3), assumed known and not in the spectrum of M(λ). Not fitted to data.
  • Time scale separation ε
    Fast-slow parameter, set to ε≪1 and taken to ε=0 in the linearized analysis. Not fitted.
  • AMOC parameter η² = 7.5
    Ratio of diffusive to advective time scale in the Stommel-Cessi model, set from [12]. Not a free parameter of the theory.
assumptions (5)
  • domain assumption Linearization of the fast subsystem along the attracting critical manifold is valid near the bifurcation threshold
    Invoked in (2.10) and discussed in (A2). The entire scaling law analysis depends on this linearization being a good approximation. Stated to weaken near the critical threshold where nonlinear terms dominate.
  • domain assumption Time-asymptotic limit (t→∞) approximates finite-time observables
    All analytic results use the time-asymptotic limit. Discussed in (A3) as an idealization. Justified under 'sufficient resilience' but the finite-time correction is not rigorously bounded.
  • domain assumption Fractional Brownian motion with additive noise is an appropriate model for non-Markovian fluctuations
    The fBm framework (Definition 2.7) is adopted as the noise model. The paper extends to fOU and red noise but the core theory is built on fBm. The Gaussian assumption is implicit.
  • standard math The critical manifold is normally hyperbolic and attracting away from the bifurcation point
    Standard fast-slow systems assumption invoked in Section 2.3. Required for the linearization and for trajectories to track the critical manifold.
  • standard math Eigenvectors of M(λ) and M(λ)^T form a biorthogonal system
    Used in Section 2.4 and throughout. Standard linear algebra result for diagonalizable matrices.

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Pith. "Pith review of Impact of spinning on the early-warning signs in non-Markovian stochastic systems." pith.science (2026). https://pith.science/paper/NYEILOFZ

@misc{pith2026260706428,
  author       = {Pith},
  title        = {Pith review of: Impact of spinning on the early-warning signs in non-Markovian stochastic systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NYEILOFZ}},
  note         = {Machine review of arXiv:2607.06428}
}
read the original abstract

We construct early-warning signals for impending critical transitions in non-Markovian systems. We analyze stochastic forcings such as fractional Brownian motion, fractional Ornstein-Uhlenbeck processes and red noise in fast-slow systems exhibiting such transitions. We show that the effectiveness of indicators such as autocovariance, autocorrelation, and spectral density depends on several properties of the underlying system. In particular, we compare the influence of the Hurst index and the bifurcation type. We prove that the rotatory dynamics associated with a Hopf bifurcation substantially alters the scaling laws of these observables. Finally, we provide practical guidelines for implementing these signals and validate them on both theoretical and applied models.

Figures

Figures reproduced from arXiv: 2607.06428 by the authors.

Figure 1
Figure 1. Structure of the paper and summary of the main results. The paper is structured as follows. Section 2 introduces the probabilistic foundation and dy￾namical setting, including the relevant statistical observables, fractional Brownian motion, and fast-slow systems with non-Markovian forcing. We also present the approximation methods and spectral notation used throughout the paper. Section 3 develops the analysis for … view at source ↗
Figure 2
Figure 2. Illustration of the portions of rays in the complex plane along which the incomplete gamma functions are studied. While the upper incomplete gamma function is well-posed only on rays with angles in − π 2 , π 2  , such as the blue solid lines, the lower incomplete gamma function does not share such a restriction. In fact, its integral can be taken along the red line in the left side of the complex plane. Nonetheless… view at source ↗
Figure 3
Figure 3. The upper part of the figure shows a stochastic process approaching a subcritical pitchfork bifurcation, solving (6.1) with ε = 10−3 , σ = 10−1 , and H = 0.75. The slow variable increases linearly toward λ ∗ = 0 with initial condition y0 = −20, and its value is encoded by the color of the arrow. The fast component (black) evolves in the direction of the arrow; its noise is generated via the Davies-Harte method and i… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: (Simulation parameters T = 211, N = 215 , samples = 20) Comparing different EWSs for the fBM noise AMOC system (6.2). In each plot we have H = 0.25 in blue, H = 0.5 in orange and H = 0.75 in green. In (a), we see the log-variance, i.e. log10 (V(0)), as the solid line a…
Figure 5
Figure 5. Figure 5: Spectral densities for H = 0.25 (a), H = 0.5 (b) and H = 0.75 (c) for (6.2). the bifurcation. Before discussing the structure of the SD, we note that due to the numerical computation with Welch’s method [13] we have a dip at the center frequency. This is caused by the …
Figure 4
Figure 4. Figure 4: Qualitatively, the EWSs behave equivalently for both systems. As the SD is also very [PITH_FULL_IMAGE:figures/full_fig_p041_4.png]
Figure 6
Figure 6. Figure 6: (Simulation parameters T = 211, N = 215 , samples = 20) Comparing different EWSs for the red noise AMOC system (6.3). In each plot we have H = 0.25 in blue, H = 0.5 in orange and H = 0.75 in green. In (a), we see the log-variance, i.e log10 (V(0)) as the solid line and…
Figure 7
Figure 7. Figure 7: (Simulation parameters T = 211, N = 215 , samples = 20) Comparing different EWSs for the fOU noise AMOC system (6.4). In each plot we have H = 0.25 in blue, H = 0.5 in orange and H = 0.75 in green. In (a), we see the log-variance log10(V(0)) as the solid line and the l…
Figure 8
Figure 8. Figure 8: Spectral densities for H = 0.25 (a), H = 0.5 (b) and H = 0.75 (c) for (6.4). 6.3 Hopf bifurcations & limit cycles We study a Hopf bifurcation in the following system driven by fBm dxt =  yt −ω0 ω0 yt  xt − |xt | 2 xt dt + [PITH_FULL_IMAGE:figures/full_fig_p043_8.png]
Figure 9
Figure 9. Figure 9: Example trajectories of x that solves (6.5) with ε = 2−5 , y0 = −1, σ1 = 0.01, σ2 = 0.05 for H = 0.25 (a), H = 0.5 (b) and H = 0.75 (c). Each axis indicates a single coordinate. fast subsystem, i.e. (6.5) with ε = 0. We start with the autocovariance, which can be seen …
Figure 10
Figure 10. Figure 10: (Simulation parameters T = 212, N = 216 , samples = 8) Plots of the different EWSs for the Hopf system driven by fBm (6.5) with σ1 = 0.01, σ2 = 0.05. Each column correspond respectively to the first component, second component and mixed component. (a)–(c): Autocovaria…
Figure 11
Figure 11. Figure 11: The first component of SD S(ω) of the Hopf system driven by fBm (6.5) for (a): H = 0.25, (b): H = 0.5 and (c): H = 0.75. After studying the Hopf system driven by a fBm, we consider the Hopf fast subsystem driven by fOU noise dxt =  λ −ω0 ω0 λ  xt − |xt | 2 xt dt + …
Figure 12
Figure 12. Figure 12: (Simulation parameters T = 212, N = 216 , samples = 8) Plots of the different EWSs for the Hopf system driven by fOU noise (6.6) with σ1 = 0.01, σ2 = 0.05 and ω0 = 1. Each column corresponds respectively to the first component, second component and mixed component. (a…
Figure 13
Figure 13. Figure 13: The first component of SD S(ω) of the Hopf system driven by fOU noise (6.6) for (a): H = 0.25, (b): H = 0.5 and (c): H = 0.75. approaching bifurcation. We omit the presentation of the red noise case, as it behaves very similarly to the fBm case. 46 [PITH_FULL_IMAGE:f…

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Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [31]

    Warning signs for non-Markovian bifurcations: colour blindness and scaling laws

    C. Kuehn, K. Lux, and A. Neamţu. “Warning signs for non-Markovian bifurcations: colour blindness and scaling laws.” In:Proc. R. Soc. A: Math. Phys. Eng.478.2259 (2022), p. 20210740

    work page 2022

  2. [1]

    Bifurcation analysis of a conceptual model for vertical mixing in the North Atlantic

    J. Bailie and B. Krauskopf. “Bifurcation analysis of a conceptual model for vertical mixing in the North Atlantic.” In:Phys. D: Nonlinear Phenom.460 (2024), p. 134077

    work page 2024

  3. [2]

    Multiparameter processes with stationary increments: Spectral representation and integration

    A. Basse-O’Connor, S.-E. Graversen, and J. Pedersen. “Multiparameter processes with stationary increments: Spectral representation and integration.” In:Electron. J. Probab. 17 (2012), no. 74, 1–21.issn: 1083-6489.doi:10 . 1214 / EJP . v17 - 2287.url:http : //ejp.ejpecp.org/article/view/2287

    work page 2012

  4. [3]

    Concentration estimates for SPDEs driven by fractional Brownian motion

    N. Berglund and A. Blessing. “Concentration estimates for SPDEs driven by fractional Brownian motion.” In:Electron. Commun. Probab.30 (2025), pp. 1–13

    work page 2025

  5. [4]

    Stochastic dynamic bifurcations and excitability

    N. Berglund and B. Gentz. “Stochastic dynamic bifurcations and excitability.” In:Stochas- tic methods in Neuroscience, C. Laing and G. Lord, (Eds.)(2008), pp. 65–93

    work page 2008

  6. [5]

    Early-Warning Signs for Bifurcation-and Noise-induced Tipping in Stochastic Partial Differential Equations

    P. Bernuzzi. “Early-Warning Signs for Bifurcation-and Noise-induced Tipping in Stochastic Partial Differential Equations.” PhD thesis. Universität München, 2026. 47

    work page 2026

  7. [6]

    Warning signs for boundary noise and their application to an ocean Boussinesq model

    P. Bernuzzi, H. A. Dijkstra, and C. Kuehn. “Warning signs for boundary noise and their application to an ocean Boussinesq model.” In:Phys. D: Nonlinear Phenom.470 (2024), p. 134391

    work page 2024

  8. [7]

    Early warning signs for SPDEs with continuous spectrum

    P. Bernuzzi, A. F. S. Düx, and C. Kuehn. “Early warning signs for SPDEs with continuous spectrum.” In:Eur. J. Appl. Math.36.4 (2025), pp. 665–704

    work page 2025

Show all 43 references
  1. [8]

    Criticalslowingdowninbifurcatingstochasticpartial differential equations with red noise

    P.Bernuzzi,C.Kuehn,andA.Morr.“Criticalslowingdowninbifurcatingstochasticpartial differential equations with red noise.” In:Partial Diff. Equ. Appl.7.1 (2026), p. 13

    2026

  2. [9]

    Biagini et al.Stochastic calculus for fractional Brownian motion and applications

    F. Biagini et al.Stochastic calculus for fractional Brownian motion and applications. Springer, 2008

    2008

  3. [10]

    Well-posedness and Hurst parameter estimation for fluid equations driven by fractional transport noise

    A. Blessing, D. Crisan, and O. Lang. “Well-posedness and Hurst parameter estimation for fluid equations driven by fractional transport noise.” In:arXiv:2604.05910(2026)

    2026 arXiv

  4. [11]

    Destabilization of Earth system tipping elements

    N. Boers et al. “Destabilization of Earth system tipping elements.” In:Nat. Geosci.18.10 (2025), pp. 949–960

    2025

  5. [12]

    A Simple Box Model of Stochastically Forced Thermohaline Flow

    P. Cessi. “A Simple Box Model of Stochastically Forced Thermohaline Flow.” In:J. Phys. Oceanogr.24.9 (1994), pp. 1911–1920.doi:10.1175/1520-0485(1994)024<1911:ASBMOS> 2.0.CO;2.url:https://journals.ametsoc.org/view/journals/phoc/24/9/1520- 0485_1994_024_1911_asbmos_2_0_co_2.xml

    1994 doi

  6. [13]

    Effective detection of early warning signal with power spectrum in climate change system

    Z. Chen et al. “Effective detection of early warning signal with power spectrum in climate change system.” In:Chaos Solit. Fractals187 (2024), p. 115409.issn: 0960-0779.doi: https://doi.org/10.1016/j.chaos.2024.115409.url:https://www.sciencedirect. com/science/article/pii/S096...

    2024 doi

  7. [14]

    Fractional Ornstein-Uhlenbeck processes

    P. Cheridito, H. Kawaguchi, and M. Maejima. “Fractional Ornstein-Uhlenbeck processes.” In:Electron. J. Probab.8.none (2003), pp. 1–14.doi:10.1214/EJP.v8-125.url:https: //doi.org/10.1214/EJP.v8-125

    2003 doi

  8. [15]

    Diffusion Properties of Small-Scale Fractional Transport Mod- els

    P. Cifani and F. Flandoli. “Diffusion Properties of Small-Scale Fractional Transport Mod- els.” In:J. Stat. Phys.192 (152 2025)

    2025

  9. [16]

    Seeking more robust early warning signals for climate tipping points: the ratio of spectra method (ROSA)

    J. J. Clarke et al. “Seeking more robust early warning signals for climate tipping points: the ratio of spectra method (ROSA).” In:Environ. Res. Lett.18.3 (Feb. 2023), p. 035006. doi:10.1088/1748-9326/acbc8d.url:https://doi.org/10.1088/1748-9326/acbc8d

    2023 doi

  10. [17]

    Early warning signals of recovery in complex systems

    C. Clements, M. McCarthy, and J. L. Blanchard. “Early warning signals of recovery in complex systems.” In:Nat. Commun.10.1681 (2019)

    2019

  11. [18]

    Da Prato and J

    G. Da Prato and J. Zabczyk.Stochastic equations in infinite dimensions. Cambridge uni- versity press, 2014

    2014

  12. [19]

    Simulation of fractional Brownian motion

    T. Dieker. “Simulation of fractional Brownian motion.” PhD thesis. Masters Thesis, De- partment of Mathematical Sciences, University of Twente, 2004

    2004

  13. [20]

    Tipping points: early warning and wishful thinking

    P. D. Ditlevsen and S. J. Johnsen. “Tipping points: early warning and wishful thinking.” In:Geophys. Res. Lett.37.19 (2010)

    2010

  14. [21]

    Semilinear stochastic equations in a Hilbert space with a fractional Brownian motion

    T. E. Duncan, B. Maslowski, and B. Pasik-Duncan. “Semilinear stochastic equations in a Hilbert space with a fractional Brownian motion.” In:SIAM J. Appl. Math.40.6 (2009), pp. 2286–2315. 48

    2009

  15. [22]

    Fractional Brownian motion and stochastic equations in Hilbert spaces

    T. E. Duncan, B. Pasik-Duncan, and B. Maslowski. “Fractional Brownian motion and stochastic equations in Hilbert spaces.” In:Stoch. Dyn.2.02 (2002), pp. 225–250

    2002

  16. [23]

    Sample paths estimates for stochastic fast-slow systems driven by fractional Brownian motion

    K. Eichinger, C. Kuehn, and A. Neamţu. “Sample paths estimates for stochastic fast-slow systems driven by fractional Brownian motion.” In:J. Stat. Phys.179.5 (2020), pp. 1222– 1266

    2020

  17. [24]

    Early warning signals from the periphery

    M. Füllsack et al. “Early warning signals from the periphery.” In:J. Comput. Soc. Sc.5 (2022), pp. 665–685

    2022

  18. [25]

    Hurst Index Estimation in Stochastic Differential Equations Driven by Fractional Brownian Motion

    J. Gairing et al. “Hurst Index Estimation in Stochastic Differential Equations Driven by Fractional Brownian Motion.” In:J. Theor. Probab.33 (3 2020), pp. 1691–1714

    2020

  19. [26]

    Guckenheimer and P

    J. Guckenheimer and P. Holmes.Nonlinear oscillations, dynamical systems, and bifurca- tions of vector fields. Springer Science & Business Media, 2013

    2013

  20. [27]

    Averaging dynamics driven by fractional Brownian motion

    M. Hairer and X.-M. Li. “Averaging dynamics driven by fractional Brownian motion.” In: Ann. Probab.48 (4 2020), pp. 1826–1860

    2020

  21. [28]

    Generating diffusions with fractional Brownian motion

    M. Hairer and X.-M. Li. “Generating diffusions with fractional Brownian motion.” In: Commun. Math. Phys.396 (2022), pp. 91–141

    2022

  22. [29]

    Kubilius, Y

    K. Kubilius, Y. Mishura, and K. Ralchenko.Parameter estimation in fractional diffusion models. Vol. 8. Springer, 2017

    2017

  23. [30]

    Kuehn et al.Multiple time scale dynamics

    C. Kuehn et al.Multiple time scale dynamics. Vol. 191. Springer, 2015

    2015

  24. [32]

    Scaling laws and warning signs for bifurcations of SPDEs

    C. Kuehn and F. Romano. “Scaling laws and warning signs for bifurcations of SPDEs.” In: Eur. J. Appl. Math.30.5 (2019), pp. 853–868

    2019

  25. [33]

    Slow-fast systems with fractional environment and dynamics

    X.-M. Li and J. Sieber. “Slow-fast systems with fractional environment and dynamics.” In: Ann. Appl. Probab.32 (5 2022), pp. 3964–4003

    2022

  26. [34]

    The influence of data gaps and outliers on resilience indicators

    T. Liu et al. “The influence of data gaps and outliers on resilience indicators.” In:arXiv preprint arXiv:2505.19034(2025)

    2025 arXiv

  27. [35]

    Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green’s Functions, and Nonlinear Fingerprints

    V. Lucarini and M. Chekroun. “Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green’s Functions, and Nonlinear Fingerprints.” In:Phys. Rev. Lett. 133.244201 (2024)

    2024

  28. [36]

    Behavior of Early Warnings near the Critical Temperature in the Two- Dimensional Ising Model

    I. Morales et al. “Behavior of Early Warnings near the Critical Temperature in the Two- Dimensional Ising Model.” In:PLos One10 (6 2015)

    2015

  29. [37]

    Detection of approaching critical transitions in natural systems driven by red noise

    A. Morr and N. Boers. “Detection of approaching critical transitions in natural systems driven by red noise.” In:Phys. Rev. X14.2 (2024), p. 021037

    2024

  30. [38]

    Red noise in continuous-time stochastic modelling

    A. Morr, D. Kreher, and N. Boers. “Red noise in continuous-time stochastic modelling.” In:R. Soc. Open Sci.12.8 (Aug. 2025), p. 250573.issn: 2054-5703.doi:10.1098/rsos. 250573. eprint:https://royalsocietypublishing.org/rsos/article- pdf/doi/10. 1098/rsos.250573/2823160/rsos.25...

    2025 doi

  31. [39]

    Nourdin.Selected Aspects of Fractional Brownian Motion

    I. Nourdin.Selected Aspects of Fractional Brownian Motion. Springer, 2012

    2012

  32. [40]

    Limit cycles as stationary states of an extended harmonic balance ansatz

    J. del Pino, J. Kosata, and O. Zilberberg. “Limit cycles as stationary states of an extended harmonic balance ansatz.” In:Phys. Rev. Research6.033180 (2024)

    2024

  33. [41]

    Integration questions related to fractional Brownian motion

    V. Pipiras and M. Taqqu. “Integration questions related to fractional Brownian motion.” In:Probab. Theory Relat. Fields118 (2000), pp. 251–291.url:https://doi.org/10. 1007/s440-000-8016-7

    2000

  34. [42]

    M. B. Priestley.The Spectral Analysis of Time Series. Vol. 151. 3. May 1988, pp. 573–574. doi:10.2307/2983035

    1988 doi

  35. [43]

    Early-warning signals for critical transitions

    M. Scheffer et al. “Early-warning signals for critical transitions.” In:Nature461 (53–59 2009). A Time-asymptotic autocovariance and autocorrelation in the one-dimensional model In this appendix, we prove the results in Section 3.1 and obtain the rate of the time-asymptotic au...

    2009

Pith tools

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