Different behaviors of diffusing diffusivity dynamics based on three different definitions of fractional Brownian motion

Aleksei V. Chechkin; Ralf Metzler; Wei Wang

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Different behaviors of diffusing diffusivity dynamics based on three different definitions of fractional Brownian motion

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Pith's one-line read Adding a fluctuating diffusivity to fractional Brownian motion yields three distinct sets of predictions: the Mandelbrot–van Ness and Riemann–Liouville forms keep an effective diffusivity equal to the mean $\langle D\rangle$, while the…

desk verdict Solid model comparison with clean derivations for two of the three FBM-DD variants; the load-bearing independence assumption should be stated explicitly. read the letter →

arxiv 2504.19190 v1 pith:BYBSC32U submitted 2025-04-27 cond-mat.stat-mech physics.bio-phq-bio.QM

Wei Wang , Aleksei V. Chechkin , Ralf Metzler This is my paper · ORCID

classification cond-mat.stat-mechphysics.bio-phq-bio.QM MSC 60G2282C31 PACS 05.40.Fb02.50.Ey

keywords fractionalBrownianmotiondiffusingdiffusivityanomalousdiffusionGaussiannoisemean-squareddisplacementnon-GaussianOrnstein-Uhlenbeckprocessyet

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 generalizes fractional Brownian motion (FBM) to heterogeneous environments by letting the diffusion coefficient itself become a random process, the square of an Ornstein–Uhlenbeck process, in each of FBM's three standard representations. It argues that the three resulting models are not equivalent: the Mandelbrot–van Ness and Riemann–Liouville versions behave as FBM with an effective diffusivity equal to the mean value $\langle D\rangle$, while the Langevin version feels the diffusivity's own correlations, producing a crossover in the mean-squared displacement (MSD) and mean-squared increment (MSI) from anomalous scaling $2\langle D\rangle t^{2H}$ to normal diffusion $2D_{\mathrm{eff}}t$ when $H<1/2$. The Riemann–Liouville version additionally has nonstationary increments, unlike the other two. These distinctions matter for experimentalists fitting single-particle tracks in viscoelastic, heterogeneous media, where the same data could be read as different transport physics depending on which FBM representation underlies the model.

What carries the argument

The central object is the squared Ornstein–Uhlenbeck process $D(t)=Y^2(t)$, with mean $\langle D\rangle=\sigma^2\tau/2$ and square-root correlation $K(\Delta)=\langle \sqrt{D(t)}\sqrt{D(t+\Delta)}\rangle$ given by Eq. (25). When inserted into the Langevin representation, $K(\Delta)$ is convolved with the fractional Gaussian noise autocovariance $\langle \xi_H^2\rangle_\Delta$; for $H<1/2$ the long-time integral converges to a linear-in-$t$ term, producing the crossover to normal diffusion. In the Mandelbrot–van Ness and Riemann–Liouville representations the diffusivity appears linearly inside the Itô integral, so under the independence assumption the average factors into $\langle D\rangle$ times the deterministic FBM kernel, and the standard $t^{2H}$ scaling survives unchanged. The large-lag expansion $K(\Delta)\sim\sigma^2\tau(\pi^{-1}+e^{-2\Delta/\tau})$ reveals the competing persistent-noise and truncated-power-law contributions that shape the LE-FBM-DD ACVF and explain the crossover.

What would settle it

Simulate the LE-FBM-DD equation with diffusivity noise that is partially correlated with the fractional Gaussian noise increments $\xi_H(t)$ and compare the MSD: the exact factorized form (38) would acquire an extra correlation term, and the clean crossover from $2\langle D\rangle t^{2H}$ to $2D_{\mathrm{eff}} t$ would shift or vanish. A cleaner laboratory test is to measure the MSI at fixed lag as a function of absolute time in a viscoelastic fluid: RL-FBM-DD predicts an aging MSI, while LE-FBM-DD and MN-FBM-DD predict a stationary one.

Watch

Extended reading notes

Core claim

On the paper's own terms, for $D(t)=Y^2(t)$ with $Y$ an Ornstein–Uhlenbeck process, the three generalized models give distinct statistical signatures. The Langevin form (LE-FBM-DD) has $\langle x^2(t)\rangle_{\mathrm{LE}} = 4\int_0^t (t-s)K(s)\langle \xi_H^2\rangle_s\,ds$, which at short times is $2\langle D\rangle t^{2H}$ and at long times $2K_{\mathrm{eff}} t^{2H}$ for $H>1/2$ but $2D_{\mathrm{eff}} t$ for $H<1/2$, with $K_{\mathrm{eff}}=\lim_{\Delta\to\infty}K(\Delta)$ and $D_{\mathrm{eff}}=2\int_0^\infty K(s)\langle \xi_H^2\rangle_s\,ds$. The Mandelbrot–van Ness form (MN-FBM-DD) gives $\langle x^2(t)\rangle_{\mathrm{MN}}=2\langle D\rangle t^{2H}$ exactly, the same as the Riemann–Liouville form (RL-FBM-DD), because the diffusivity factors out of the Itô integrals under the equilibrium and independence assumptions. The increments of MN-FBM-DD and LE-FBM-DD are stationary; those of RL-FBM-DD are not, with MSI $4H\langle D\rangle[I_H(t/\Delta)+1/(2H)]\Delta^{2H}$. The autocovariance function (ACVF) of LE-FBM-DD carries the diffusivity correlation $K(\Delta)$ multiplied by the fractional-noise autocovariance, while the other two carry only $\langle D\rangle$ times their usual noise autocovariance. All three displacement PDFs cross over from a short-time non-Gaussian (Bessel-function) form to a long-time Gaussian form.

Load-bearing premise

The derivations assume the random diffusivity $D(t)$ is statistically independent of the Brownian motion $B(s)$ driving the fractional Gaussian noise (and, for the Mandelbrot–van Ness form, that the diffusivity has already reached equilibrium for all times $s\le 0$), because the clean factorizations into $\langle D\rangle$ and $K(\Delta)$ require the averages to separate; if the diffusivity were correlated with the noise, every predicted scaling and crossover would change.

Editorial extensions

If this is right

For $H<1/2$, the LE-FBM-DD model predicts that a tracer's long-time MSD becomes linear in time, so fitting only long-time trajectories would make the motion look Brownian even though the process is fractional at short times.
The MSI and ACVF are stationary for LE-FBM-DD and MN-FBM-DD but nonstationary for RL-FBM-DD, providing experimental signatures in increment statistics that do not require measuring the MSD.
All three DD-generalized models predict a crossover of the displacement PDF from a short-time non-Gaussian shape to a long-time Gaussian, with the crossover scale set by the diffusivity correlation time $\tau$.
MN-FBM-DD and RL-FBM-DD both yield $\langle x^2(t)\rangle=2\langle D\rangle t^{2H}$ for all $H$ in $(0,1)$, meaning the long-time MSD alone cannot distinguish those two representations.

Reading between the lines

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

Because the factorization to $\langle D\rangle$ relies on $D(t)$ being independent of the driving noise, a testable extension is to couple $D(t)$ weakly to the particle position: the clean $t^{2H}$ scaling should break first in the MN- and RL-forms, while the LE-form's crossover would shift.
The same distinction should apply to multifractional Brownian motion with a time-dependent Hurst exponent: representations based on the Langevin form should show diffusivity-correlation crossovers in addition to Hurst-exponent aging effects.
Single-particle tracking pipelines that treat the three FBM representations as interchangeable could misclassify subdiffusive trajectories, because the fitted long-time exponent would be $1/2$ under LE-FBM-DD but $2H$ under MN-FBM-DD for the same underlying physics.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Here is my take on Wang et al. (arXiv:2504.19190). The paper asks a good question: when you add diffusing diffusivity (DD) to fractional Brownian motion, does it matter which representation of FBM you start from? The answer is yes, and the paper shows it carefully. The MN-FBM-DD and RL-FBM-DD results are new: their MSDs, MSIs, and ACVFs just pick up an effective diffusivity equal to the mean . In contrast, LE-FBM-DD shows a crossover in the MSD/MSI for H<1/2, which was already reported in the authors' earlier work (ref 72). The systematic comparison and the summary table are the main contributions.

The derivations for MN- and RL-FBM-DD are straightforward and exact, relying only on Itô isometry and stationarity of the OU-squared diffusivity. The simulations agree with the analytic expressions. The paper is clearly written and honest about what is taken from prior work.

The main soft spot is a missing hypothesis. The derivations factor E[D(s)] out of Itô integrals and factor the correlation of sqrt(D) out of the double integral. This only holds if the diffusivity process D(t) is independent of the Brownian motion B(s) driving the FBM integrals. That independence is standard in the field, but it is load-bearing and never stated. The same applies to the equilibrium assumption for D(s) on negative times in the MN representation. This is not an error, but it should be an explicit model hypothesis. A referee should ask for that.

Minor issues: Eq. (55) for the LE-FBM-DD ACVF appears without derivation, though the form is plausible. The crossover results for LE-FBM-DD are quoted from ref 72, so the genuinely new material is narrower than the title suggests. Also, the paper does not discuss what happens if D(t) and B(s) are correlated, which would change all three predictions, but that is beyond the intended model.

Who is this for? Experimentalists analyzing single-particle tracking in viscoelastic, heterogeneous media, and theorists building model libraries for anomalous diffusion. It deserves a serious referee. I would accept it for peer review with a request to make the independence assumption explicit and to add a sentence on its physical meaning.

Best.

Referee Report

1 major / 3 minor

Summary. The manuscript generalizes three equivalent representations of fractional Brownian motion (Langevin equation driven by fractional Gaussian noise, LE-FBM; Mandelbrot–van Ness integral, MN-FBM; and Riemann–Liouville integral, RL-FBM) by replacing the constant diffusivity with a random diffusivity D(t) that evolves as the square of an Ornstein–Uhlenbeck process. The authors derive the mean-squared displacement (MSD), mean-squared increment (MSI), autocovariance function of increments (ACVF), and probability density function for the resulting FBM-DD models. The central claim is that although MN-FBM and LE-FBM are equivalent for constant diffusivity, their diffusing-diffusivity counterparts behave differently: MN-FBM-DD and RL-FBM-DD both give MSD = 2<D> t^{2H} with the increment-stationarity properties of the underlying FBM, whereas LE-FBM-DD exhibits an unexpected crossover to Brownian diffusion for H<1/2 and to a rescaled anomalous scaling for H>1/2, driven by correlations of the random diffusivity. Simulations are presented for the MSD, MSI, ACVF, and PDF and are reported to agree with the analytical predictions.

Significance. If the results hold, the paper is significant because it demonstrates that the three standard representations of fractional Brownian motion, equivalent for constant diffusivity, are not equivalent when coupled to a random diffusivity. It provides a systematic comparison, summarized in Table I, and makes falsifiable predictions, such as the subdiffusive crossover to normal diffusion in LE-FBM-DD, that could be used to discriminate between competing FBM-DD models in single-particle-tracking experiments on viscoelastic, heterogeneous media. The derivations for the MN- and RL-FBM-DD models are exact and transparent, relying only on the Itô isometry and the stationarity of the squared-OU diffusivity, and the simulation methodology is described in enough detail to be reproduced.

major comments (1)

[Section III.B, Eqs. (38) and (55)] The model definitions do not specify the joint law of the OU noise η(t) and the Brownian motion or fractional Gaussian noise that drives the particle. The derivation of the LE-FBM-DD MSD in Eq. (38) and the ACVF in Eq. (55) requires that D(t), equivalently η(t), be independent of ξ_H(t); if this independence fails, cross-correlation terms appear and the factorizations in Eqs. (38) and (55) are invalid. Since the predicted crossover in Eq. (41) is the paper's headline result, the independence hypothesis must be stated explicitly as part of the model definition. Note that the MN- and RL-FBM-DD results do not require this independence, only the Itô isometry and stationarity of D(t), so the missing hypothesis is specific to the LE-FBM-DD claims.

minor comments (3)

[Section III.A.2, text near Eq. (9)] The sentence 'MN-FBM and LE-FBM are equivalent in the sense that they exhibiting the same MSD...' contains a grammatical error; 'they exhibiting' should be 'they exhibit'.
[Eq. (56)] The large-Δ expansion of K(Δ) from Eq. (25) gives σ²τ/π [1 + (1/2)e^{-2Δ/τ}] to leading order, so the exponential term in Eq. (56) should carry a prefactor 1/(2π); as written, the coefficient of the exponential term is too large by a factor of 2π.
[Section V.C, Eq. (55)] A brief derivation of the LE-FBM-DD ACVF would be helpful, as this result is used in interpreting the MSD crossover and is not explicitly derived in the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the MSD/MSI/ACVF results follow from the model definitions via standard stochastic calculus, with no fitted parameter relabeled as a prediction.

full rationale

The paper defines three DD models by inserting D(t)=Y^2(t) into the three FBM representations, and the central results are direct consequences of these definitions plus standard Itô-isometry and stationarity arguments. For MN-FBM-DD, Eq. (43) factors E[D(s)] = <D> out of the stochastic integrals under the stated equilibrium and independence assumptions, and the remaining bracket evaluates to 1/V_H by the same normalization used to define MN-FBM; Eq. (44) is therefore a calculation, not a fit. RL-FBM-DD yields the same MSD by the Itô isometry, again without any tuning. The LE-FBM-DD MSD, Eq. (38), is quoted from the authors' earlier paper [72], but that is an independent, parameter-free analytic result and it is numerically reproduced in Fig. 1, so citing it is legitimate prior support rather than circular reasoning. The ACVF factorizations (55) and (59) rely on the explicit modeling assumption that the OU-squared diffusivity is independent of the driving Gaussian noise; that is a model hypothesis, not an output disguised as an input. No equation in the paper is equivalent to its own input by construction, no fitted parameter is renamed as a prediction, and no uniqueness claim is imported from the authors' earlier work. The main caveat—the implicit independence of D(t) from the Brownian driver—is a correctness and scope condition, not a circularity, and does not raise the circularity score.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model depends on three parameters (H, tau, sigma) that define the stochastic processes but are not fitted to data. The derivations assume independence of the diffusivity and the driving noise, stationarity of the OU-squared process, and standard FBM background. For MN-FBM-DD, an additional modeling choice is that the diffusivity is in equilibrium for negative times. No new physical entities are introduced.

free parameters (3)

H (Hurst exponent) = 0.2 and 0.8 in simulations
Model parameter of FBM, chosen to illustrate subdiffusion and superdiffusion. The analytical results hold for general 0<H<1; no fitting to data.
tau (OU correlation time) = 1 in simulations
Sets the correlation time of the diffusivity; the crossover time is proportional to tau. Chosen by hand; results are for general tau.
sigma (OU noise intensity) = 1 in simulations
Controls the amplitude of diffusivity fluctuations; effective diffusivity <D> = sigma^2 tau / 2. Chosen by hand; general results valid.

assumptions (5)

domain assumption D(t) is statistically independent of the Brownian motion B(s) driving the FBM representations.
Used throughout the derivations of MSD, MSI, ACVF (e.g., Eq. (43) replaces <D(s) dB(s)^2> with <D(s)> ds). This is standard in diffusing diffusivity models but is a physical assumption about the noise sources.
standard math Fractional Gaussian noise with ACVF given by Eq. (8) is a well-defined stationary Gaussian process for 0<H<1.
Background for LE-FBM and the equivalence of LE-FBM and MN-FBM. Standard in the FBM literature.
domain assumption The OU-squared process D(t)=Y^2(t) with equilibrium initial condition (23) is stationary with mean <D>=sigma^2 tau/2 and correlation K(Delta) given by Eq. (25).
Standard OU calculus; the explicit K(Delta) expression is from refs 61 and 72.
ad hoc to paper For the MN-FBM-DD model, the diffusivity D(s) is at equilibrium for all s<=0, including negative times.
This is a modeling choice specific to the Weyl representation, stated in Sec. III.B. It is required for the second integral over (-inf,0] to be stationary and to yield the <D> factorization. If D(s) for s<=0 were not stationary, the MN-FBM-DD MSD would differ.
domain assumption The LE-FBM-DD SDE (29) is interpreted formally, and the MSD expression (38) from ref 72 is valid.
fGn is not a semimartingale for H != 1/2; the Langevin equation is treated as a formal stochastic equation in the physics literature. The result (38) is cited from prior work and confirmed by simulation.

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Pith. "Pith review of Different behaviors of diffusing diffusivity dynamics based on three different definitions of fractional Brownian motion." pith.science (2026). https://pith.science/paper/BYBSC32U

@misc{pith2026250419190,
  author       = {Pith},
  title        = {Pith review of: Different behaviors of diffusing diffusivity dynamics based on three different definitions of fractional Brownian motion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BYBSC32U}},
  note         = {Machine review of arXiv:2504.19190}
}

read the original abstract

The effects of a "diffusing diffusivity" (DD), a stochastically time-varying diffusion coefficient, are explored within the frameworks of three different forms of fractional Brownian motion (FBM): (i) the Langevin equation driven by fractional Gaussian noise (LE-FBM), (ii) the Weyl integral representation introduced by Mandelbrot and van Ness (MN-FBM), and (iii) the Riemann-Liouville fractional integral representation (RL-FBM) due to L{\'e}vy. The statistical properties of the three FBM-generalized DD models are examined, including the mean-squared displacement (MSD), mean-squared increment (MSI), autocovariance function (ACVF) of increments, and the probability density function (PDF). Despite the long-believed equivalence of MN-FBM and LE-FBM, their corresponding FBM-DD models exhibit distinct behavior in terms of the MSD and MSI. In the MN-FBM-DD model, the statistical characteristics directly reflect an effective diffusivity equal to its mean value. In contrast, in LE-FBM-DD, correlations in the random diffusivity give rise to an unexpected crossover behavior in both MSD and MSI. We also find that the MSI and ACVF are nonstationary in RL-FBM-DD but stationary in the other two DD models. All DD models display a crossover from a short-time non-Gaussian PDF to a long-time Gaussian PDF. Our findings offer guidance for experimentalists in selecting appropriate FBM-generalized models to describe viscoelastic yet non-Gaussian dynamics in bio- and soft-matter systems with heterogeneous environments.

Figures

Figures reproduced from arXiv: 2504.19190 by the authors.

**Figure 2.** FIG. 2: Simulations (circles, triangles, rectangles) for t [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3: Simulations (circles, triangles, rectangles) for t [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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**Figure 4.** Figure 4: FIG. 4: Simulations (circles, triangles, rectangles) for t [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Quantum Brownian transport in a correlated Gaussian force
cond-mat.stat-mech 2026-07 reject novelty 4.0 of 10

A no-dissipation Caldeira-Leggett model is claimed to give bath-particle mean-square displacement ~t^5 and velocity ~t^3, but the derivation is internally inconsistent.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

We then proceed to generalize these FBM processes by incorporating the DD dynamics of their dif- fusivity, modeled as the square of an Ornstein-Uhlenbeck process

of the MSD in their original formulation with constant parameters in uncon- ﬁned space. We then proceed to generalize these FBM processes by incorporating the DD dynamics of their dif- fusivity, modeled as the square of an Ornstein-Uhlenbeck process

[2] [2]

Mandelbrot and van-Ness deﬁnition of FBM The most widely used representation of FBM, intro- duced by Mandelbrot and van-Ness (MN-FBM) [ 37] and often referred to as "FBM I" in literature [ 82], is deﬁned for the Hurst exponent 0<H < 1 in the form xMN(t) = √ 2DVH { ∫ t 0 (t −s)H−1/ 2dB(s) + ∫ 0 −∞ [ (t −s)H−1/ 2 − (−s)H−1/ 2 ] dB(s) } , (5) whereB(t) denot...

[3] [3]

Here the driving fractional Gaussian noise has zero mean and ACVF ⟨ξ2 H ⟩∆ = ⟨ξH (t + ∆) ξH (t)⟩ = 1 2δ2 ( |∆ + δ|2H + |∆ −δ|2H − 2∆ 2H)

Langevin equation formulation of FBM An alternative formulation of MN-FBM, especially widely used in physics literature, is deﬁned through the overdamped Langevin equation, a process we refer to as LE-FBM [ 37, 84], dxLE(t) dt = √ 2DξH (t), (7) for 0 < H ≤ 1. Here the driving fractional Gaussian noise has zero mean and ACVF ⟨ξ2 H ⟩∆ = ⟨ξH (t + ∆) ξH (t)⟩ ...

[4] [4]

It is expressed as x(t) = ∫ t 0 √ 4DH (t −s)H−1/ 2dB(s), (12) for H >0 RL-FBM shares the same MSD with MN-FBM and LE-FBM, ⟨x2(∆) ⟩RL = 2D∆ 2H

Riemann-Liouville formulation of FBM Recently, growing attention has been paid to an al- ternative deﬁnition of FBM introduced by Lévy [ 37, 85], which is based on the Riemann-Liouville fractional inte- gral (RL-FBM) and referred to as the "FBM II" [ 82]. It is expressed as x(t) = ∫ t 0 √ 4DH (t −s)H−1/ 2dB(s), (12) for H >0 RL-FBM shares the same MSD wit...

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The generalized diﬀusion coeﬃcients are displayed in the inset

of the LE-FBM-DD model ( 29) is depicted by the red solid curve. The generalized diﬀusion coeﬃcients are displayed in the inset. In panel (a), the MSD of all mod- els show the same scaling, and the generalized diﬀusion coef - ﬁcient ⟨x2(t)⟩/t 2H of the LE-FBM-DD has a crossover from 2⟨D⟩ = 1 to 2Keﬀ = 2/π , represented by dash-dotted lines. In panel (b), ...

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