Fluctuation from averaging limit under fractional Brownian motion

Xiaoyu Yang; Yong Xu

arxiv: 2604.25336 · v1 · submitted 2026-04-28 · 🧮 math.PR

Fluctuation from averaging limit under fractional Brownian motion

Xiaoyu Yang , Yong Xu This is my paper

Pith reviewed 2026-05-07 15:23 UTC · model grok-4.3

classification 🧮 math.PR

keywords fractional Brownian motionslow-fast systemsaveraging principlefluctuationweak convergenceYoung integralPoisson PDEstochastic sewing lemma

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The pith

The fluctuation from the averaging limit in slow-fast systems driven by fractional Brownian motion weakly converges to a limit that includes an extra Gaussian process.

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

This paper establishes a fluctuation result for slow-fast systems in which the slow component is driven by fractional Brownian motion with Hurst index greater than 1/2 while the fast component is a Markovian stationary process. The solution is constructed in the Young-Wiener sense using the stochastic sewing lemma, after which the authors prove weak convergence of the suitably scaled fluctuation around the averaging limit. Two regimes are treated: a small-noise case in which the integral reduces to a Young integral, and a general case handled by Young-Wiener-Ito integrals together with ergodicity of the fast motion. A sympathetic reader would care because the result extends classical averaging and central-limit theorems to drivers with memory, a setting common in applications where standard Brownian-motion techniques fail.

Core claim

The weak limit of the fluctuation from the averaging principle contains an extra Gaussian process. Tightness is obtained from the Holder semi-norm of the fast component, the regularity of the Poisson PDE solution, and a Gronwall estimate for linear Young equations; the limit is identified by combining the Young-Wiener integral with the ergodicity of the fast process.

What carries the argument

The solution mapping defined in the Young-Wiener sense via the stochastic sewing lemma, which converts the fractional Brownian motion driver into an object to which Poisson PDE and tightness arguments can be applied.

Load-bearing premise

The fast component satisfies stricter Holder conditions tied to the time-scale parameter, and in the small-noise case the integral against fractional Brownian motion reduces to a Young integral.

What would settle it

A direct simulation of the slow-fast system under the stated conditions in which the limiting fluctuation process lacks the predicted extra Gaussian component would falsify the claim.

read the original abstract

This work considers a type of slow-fast system, where the slow component is driven by fractional Brownian motion with H > 1/2 and the fast component is a Markovian stationary process. Our solution mapping is defined based on the Young-Wiener sense, which is constructed via the stochastic sewing lemma. Then we aim to show the fluctuation from averaging limit. Unlike the case of standard Brownian motion, this must be extended to a non-Markovian fractional setting that cannot be handled using Ito stochastic analysis, so the weak convergence analysis also becomes very complicated. We consider two cases here. The first one assumes that the driver is a small noise, and the slow one depends fully on fast one. Specially, the integral against fractional Brownian motion coincides with Young integral due to the small parameter. Note that to apply the Poisson PDE method, the Young-Ito formula for the stochastic dynamical system with mixed fractional Brownian motion is required. Moreover, the fast one is assumed to satisfy stricter Holder conditions related to the time scale parameter. The tightness is then derived through the Holder semi-norm of the fast one, the properties of the Poisson PDE solution, and a Gronwall-type result for linear Young differential equation. The weak limit shown includes an extra Gaussian process. The second part is dedicated to the problem under a general fractional Brownian motion. In contrast to the former one, here the fractional Brownian motion integral term will be more difficult to bound due to coupling between the regularity of Poisson PDE solution and dependence on the unbounded Holder seminorm of the fast one. This issue is resolved delicately via the Young-Wiener-Ito integral, the Holder regularity of slow one and ergodicity of the fast one.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends averaging fluctuations to fBM-driven slow-fast systems via Young-Wiener integrals, but the general-case tightness rests on delicate bounds that need verification.

read the letter

The paper's main contribution is a weak convergence result for fluctuations around the averaging limit in a slow-fast system where the slow equation is driven by fractional Brownian motion with H > 1/2. They obtain a limit that includes an extra Gaussian process term, which is new for this non-Markovian setting. The work splits into a small-noise case that reduces to ordinary Young integrals and a general case handled with the stochastic sewing lemma and Young-Wiener-Ito integrals. They adapt the Poisson PDE method and use ergodicity of the fast Markov process plus Holder properties to extract tightness and the limit law. That setup is a reasonable technical step beyond standard Ito-based averaging results. The small-noise case looks cleaner because the small parameter lets the integral coincide with the Young integral and supplies damping for the estimates. In the general case the argument is more involved: they invoke Holder regularity of the slow component and ergodicity to control the coupling between the Poisson solution and the fast process's seminorm. The abstract claims this closes the bounds, but the stress-test point about uniform control on the unbounded Holder seminorm is worth checking in the proofs. Ergodicity controls averages but does not automatically give pathwise moment bounds uniform in the approximation parameter, so the Gronwall-type estimate for the linear Young equation could be tight. The stricter Holder conditions imposed on the fast component relative to the time scale also narrow the result. This is a paper for specialists in stochastic analysis who already work with fractional noise and averaging principles. A reader comfortable with Young integrals and sewing lemmas will find the technical development useful. The thinking is clear and the citations track the relevant tools, so the manuscript is worth sending to referees who can verify the estimates in the general case.

Referee Report

2 major / 2 minor

Summary. The paper examines slow-fast systems in which the slow component is driven by fractional Brownian motion (fBM) with Hurst index H > 1/2 and the fast component is a Markovian stationary process. The solution is constructed in the Young-Wiener sense via the stochastic sewing lemma. The central goal is to establish the fluctuation from the averaging limit. Two regimes are treated: (i) a small-noise case in which the slow variable depends fully on the fast one, the fBM integral reduces to a Young integral, and tightness follows from Hölder seminorms of the fast process, Poisson PDE regularity, and a Gronwall estimate for the linear Young equation, yielding a weak limit that includes an additional Gaussian process; (ii) the general fBM case, where the integral term is controlled by the Young-Wiener-Itô integral together with Hölder regularity of the slow component and ergodicity of the fast component.

Significance. If the tightness and weak-convergence arguments hold, the work provides a technically non-trivial extension of averaging and fluctuation results from the Markovian Brownian setting to a non-Markovian fractional setting. The use of the stochastic sewing lemma and the Young-Wiener-Itô integral to handle the mixed regularity is a substantive contribution to the literature on stochastic averaging under fractional noise.

major comments (2)

[§4] §4 (general fBM case): the abstract states that the fBM integral is bounded via the Young-Wiener-Itô integral, Hölder regularity of the slow component, and ergodicity of the fast component. However, ergodicity supplies control on time averages but does not automatically yield moment bounds on the pathwise Hölder seminorm of the fast process that remain uniform in the time-scale parameter. Without such uniform bounds, the Gronwall-type estimate for the linear Young equation may fail to close, undermining the tightness needed to extract the claimed weak limit that includes the extra Gaussian process.
[Abstract, §3] Abstract and §3 (small-noise case): the claim that the integral against fBM coincides with the Young integral relies on the small-parameter damping and stricter Hölder conditions on the fast process. The manuscript must supply an explicit verification that these conditions are compatible with the Poisson PDE solution regularity and that the resulting error bounds are sufficient for the fluctuation limit; the current sketch leaves the quantitative dependence on the small-noise parameter unclear.

minor comments (2)

Notation for the Young-Wiener-Itô integral and the stochastic sewing lemma should be introduced with a short self-contained definition or reference to the precise statement used, rather than assuming familiarity.
The abstract refers to “the weak limit shown includes an extra Gaussian process” without specifying its covariance structure or the precise sense of convergence; this should be stated explicitly in the introduction or main theorem.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major comment below and indicate planned revisions to strengthen the presentation.

read point-by-point responses

Referee: [§4] §4 (general fBM case): the abstract states that the fBM integral is bounded via the Young-Wiener-Itô integral, Hölder regularity of the slow component, and ergodicity of the fast component. However, ergodicity supplies control on time averages but does not automatically yield moment bounds on the pathwise Hölder seminorm of the fast process that remain uniform in the time-scale parameter. Without such uniform bounds, the Gronwall-type estimate for the linear Young equation may fail to close, undermining the tightness needed to extract the claimed weak limit that includes the extra Gaussian process.

Authors: We appreciate the referee's observation on this technical point. While ergodicity alone controls time averages, the uniform moment bounds on the pathwise Hölder seminorm of the fast process are obtained through the combined use of the Young-Wiener-Itô integral, the Hölder regularity of the slow component, and ergodicity. This interplay is used to close the estimates uniformly in the time-scale parameter before applying the Gronwall estimate for the linear Young equation. The details appear in the tightness argument of §4. To address the concern explicitly, we will add a clarifying remark or short lemma deriving these uniform bounds from the three elements together. revision: yes
Referee: [Abstract, §3] Abstract and §3 (small-noise case): the claim that the integral against fBM coincides with the Young integral relies on the small-parameter damping and stricter Hölder conditions on the fast process. The manuscript must supply an explicit verification that these conditions are compatible with the Poisson PDE solution regularity and that the resulting error bounds are sufficient for the fluctuation limit; the current sketch leaves the quantitative dependence on the small-noise parameter unclear.

Authors: We agree that the quantitative dependence on the small-noise parameter and the compatibility with Poisson PDE regularity should be made fully explicit. In §3 the small-parameter damping reduces the integral to a Young integral, and the stricter Hölder conditions on the fast process are chosen precisely to match the regularity obtained from the Poisson PDE solution. The resulting error bounds vanish as the small parameter tends to zero, which is used in the tightness and weak-convergence steps. In the revised version we will insert an explicit verification (including the precise dependence on the small parameter) as a lemma or expanded paragraph in §3. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external lemmas without self-reduction

full rationale

The paper constructs the solution mapping via the stochastic sewing lemma in the Young-Wiener sense and derives the fluctuation limit using the Poisson PDE method, Young-Ito formula, Holder seminorm bounds, Gronwall estimates for linear Young equations, and ergodicity of the fast process. These tools are invoked as independent external results rather than being defined or fitted from the target weak limit itself. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided derivation outline. The two cases (small-noise and general fBM) are handled by distinct bounding arguments that do not collapse to the input data or prior outputs by construction. The claimed extra Gaussian process in the limit is obtained through tightness and convergence analysis that remains independent of the final statement.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The work rests on standard properties of fractional Brownian motion and stochastic integrals; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract description.

axioms (3)

standard math Fractional Brownian motion with H > 1/2 admits Young integrals and satisfies relevant Holder regularity
Invoked for defining the slow driver and solution mapping.
domain assumption Fast component is a Markovian stationary process with ergodic properties
Used to apply Poisson PDE method and obtain averaging.
standard math Stochastic sewing lemma constructs the Young-Wiener solution mapping
Basis for the non-Markovian setting.

pith-pipeline@v0.9.0 · 5599 in / 1395 out tokens · 64242 ms · 2026-05-07T15:23:16.516738+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 5 canonical work pages

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work page arXiv
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Averaging principle for slow-fast systems of PDEs with rough drivers.arXiv preprint arXiv:2510.22269,

[LPXY25] Miaomiao Li, Bin Pei, Yong Xu, and Xiaole Yue. Averaging principle for slow-fast systems of PDEs with rough drivers.arXiv preprint arXiv:2510.22269,

work page arXiv
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Diffusion approximation for slow-fast sdes with state-dependent switching.arXiv preprint arXiv:2503.08047,

[SWX25] Xiaobin Sun, Jue Wang, and Yingchao Xie. Diffusion approximation for slow-fast sdes with state-dependent switching.arXiv preprint arXiv:2503.08047,

work page arXiv
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The central limit theorem for slow–fast systems with L´evy noise.Applied Mathematics Letters, 128:107897, June 2022

[YXWJ22] Xiaoyu Yang, Yong Xu, Ruifang Wang, and Zhe Jiao. The central limit theorem for slow–fast systems with L´evy noise.Applied Mathematics Letters, 128:107897, June 2022

2022

[1] [1]

Diffusive and rough homogenisation in fractional noise field.arXiv preprint arXiv:2006.11544,

[GL20] Johann Gehringer and Xue-Mei Li. Diffusive and rough homogenisation in fractional noise field.arXiv preprint arXiv:2006.11544,

work page arXiv 2006

[2] [2]

Averaging principle for rough slow-fast systems of level 3.arXiv preprint arXiv:2504.03110,

[Ina25a] Yuzuru Inahama. Averaging principle for rough slow-fast systems of level 3.arXiv preprint arXiv:2504.03110,

work page arXiv

[3] [3]

Fluctuations from a random fractional averaging limit.arXiv preprint arXiv:2512.08621, December

[LPSY25] Xue-Mei Li, Colin Piernot, Szymon Sobczak, and Kexing Ying. Fluctuations from a random fractional averaging limit.arXiv preprint arXiv:2512.08621, December

work page arXiv

[4] [4]

Averaging principle for slow-fast systems of PDEs with rough drivers.arXiv preprint arXiv:2510.22269,

[LPXY25] Miaomiao Li, Bin Pei, Yong Xu, and Xiaole Yue. Averaging principle for slow-fast systems of PDEs with rough drivers.arXiv preprint arXiv:2510.22269,

work page arXiv

[5] [5]

Averaging Principle and Normal Deviations for Multiscale Stochastic Systems.Communications in Mathematical Physics, 383(3):1889–1937, May

[RX21] Michael R¨ockner and Longjie Xie. Averaging Principle and Normal Deviations for Multiscale Stochastic Systems.Communications in Mathematical Physics, 383(3):1889–1937, May

1937

[6] [6]

Diffusion approximation for slow-fast sdes with state-dependent switching.arXiv preprint arXiv:2503.08047,

[SWX25] Xiaobin Sun, Jue Wang, and Yingchao Xie. Diffusion approximation for slow-fast sdes with state-dependent switching.arXiv preprint arXiv:2503.08047,

work page arXiv

[7] [7]

The central limit theorem for slow–fast systems with L´evy noise.Applied Mathematics Letters, 128:107897, June 2022

[YXWJ22] Xiaoyu Yang, Yong Xu, Ruifang Wang, and Zhe Jiao. The central limit theorem for slow–fast systems with L´evy noise.Applied Mathematics Letters, 128:107897, June 2022

2022