Diffusion properties of small-scale fractional transport models

Franco Flandoli; Paolo Cifani

REVIEW 3 major objections 5 minor 25 references

Diffusion properties of small-scale fractional transport models

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 Spatial complexity restores ordinary Brownian diffusion to particles advected by persistent fractional noises, with the Hurst exponent surviving only in the diffusion coefficient.

desk verdict An honest numerical study of a plausible effect: spatial mixing erasing fBm memory, presented as conjecture with second-moment evidence, and that is exactly what it is. read the letter →

arxiv 2506.08068 v2 pith:CIOFI6KJ submitted 2025-06-09 cond-mat.stat-mech

Paolo Cifani , Franco Flandoli This is my paper

classification cond-mat.stat-mech MSC 60G2260H1576F2582C31 PACS 05.40.-a47.27.-i

keywords fractionalBrownianmotionstochastictransportpassivetracerdiffusionHurstexponentanomalousturbulentOrnstein-Uhlenbeckapproximationlimit

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 asks whether particles advected by a velocity field whose independent Fourier modes are each driven by a persistent fractional Gaussian noise ($H>1/2$) can nevertheless diffuse like an ordinary Brownian motion. The authors' central claim is that they do: once the spatial structure of the flow is made small-scale and spatially complex, the long-range temporal memory of the noise is lost, and the tracer position converges in law to a two-dimensional Brownian motion, with the Hurst exponent $H$ surviving only inside the diffusion coefficient. The evidence is a scaling conjecture (Theorem 1) linking the noise relaxation time $\tau_\eta$ to the spatial scale $\eta$, combined with numerical simulations showing a crossover in the mean-square displacement from slope $2H$ at short times to slope $1$ at long times. If correct, the result isolates a concrete mechanism—mixing across many independent spatial modes—that restores independent increments without any additional white-noise assumption.

What carries the argument

The load-bearing object is the stochastic velocity field $u(x,t)=u\,C(\eta,\tau,H)\sum_{k\in K_\eta}\sigma_k(x)\,dB^{H,k}_t/dt$, where $B^{H,k}_t$ are independent fractional Brownian motions, the $\sigma_k$ are divergence-free spatial modes of wavelength $\eta$, and $C(\eta,\tau,H)=\tau^{1-H}\sqrt{2}/\sqrt{\Gamma(2H+1)}/C_\eta$ with $C_\eta$ fixed by the average kinetic energy. This normalization puts different values of $H$, $\eta$, and $\tau$ on the same energetic footing, so the comparison between models is meaningful. The argument carrying the Brownian claim is the switching heuristic of Appendix C: a tracer is influenced mostly by one Fourier component at a time; after a time $t_\eta$ spent travelling a distance of order $\lambda\eta$, it moves into the influence zone of another component, and successive displacements become approximately independent because the components are driven by independent noises. This reduces the long-time motion to a random walk with step $\lambda\eta$, from which the diffusion coefficient follows.

What would settle it

Measure the mean-square displacement for the test case at fixed $\tau$ across several values of $\eta$ in the Brownian regime and plot $\sigma_H$ against $\eta$; formula (20) predicts the exponent $1-1/(2H)$ (about $0.286$ for $H=0.7$), so a different exponent would refute the diffusion-coefficient derivation. Alternatively, record which Fourier mode contributes most to the force on a tracer over time and check that the waiting times between mode changes have mean of order $t_\eta$ and are approximately independent.

Watch

Extended reading notes

Core claim

The paper's central claim is that spatial complexity erases time memory in fractional stochastic transport. For the test case—a divergence-free velocity field built from sinusoids of wavelengths comparable to $\eta$ and driven by independent fractional noises of Hurst index $H>1/2$—the Lagrangian tracer $X_t^0$ behaves, after a short transient, as a Brownian motion. Concretely, $\mathbb{E}|X_t^0|^2$ scales as $t^{2H}$ for times short compared with the time needed to travel a distance of order $\eta$, and then crosses to the linear-in-$t$ scaling of ordinary diffusion. The paper formulates this as Theorem 1: taking $\tau=\tau_\eta=C\eta^{(1-2H)/(1-H)}$, the processes $X_t^0$ converge in law to a two-dimensional Brownian motion as $\eta\to0$. The Hurst exponent does not disappear entirely; it enters the diffusion coefficient through the approximate formula $\sigma_H\sim\left(2/\Gamma(2H+1)\right)^{1/(4H)}u^{1/(2H)}\tau_\eta^{1/(2H)-1/2}\lambda^{1-1/(2H)}\eta^{1-1/(2H)}$, with a fitted, non-universal constant $\lambda$. The paper presents Theorem 1 as a conjecture rather than a proof, supported by the numerical verification and the heuristic argument of Appendix C.

Load-bearing premise

The quantitative claim rests on the heuristic that a tracer is dominated by one wave component at a time and hops to another after travelling a distance of order $\eta$, so successive displacements become approximately independent; if this switching picture fails, the predicted scaling of the diffusion coefficient loses its support.

Editorial extensions

If this is right

For times much larger than the crossover $t_*$, passive-scalar transport in this class of fields is governed by an ordinary diffusion equation with an $H$-dependent effective diffusivity; fractional scaling survives only in the transient.
The mean-square displacement gives two independent reads on the noise: the short-time slope $2H$ exposes the Hurst exponent, while the long-time slope $1$ and the level of the plateau determine the renormalized diffusion coefficient.
Under the scaling $\tau_\eta=C\eta^{(1-2H)/(1-H)}$, the Brownian limit is expected as $\eta\to0$, and for finite $\eta$ the Brownian description is approximate, improving as the spatial scale shrinks.
A swarm of particles initialized in a small cluster keeps its shape until $t\sim t_*$, then loses all initial structure and disperses with variance growing linearly in time; the long-time variance is larger for larger $H$.

Reading between the lines

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

This suggests the same mechanism should operate in three-dimensional divergence-free fields assembled from random spherical modes, with $\eta$ identified with the dominant wavelength; the predicted scaling of $\sigma_H$ could be tested in that setting.
A theory that determines the fitted constant $\lambda$ from the spectrum of the modes would remove the only free parameter; the paper's two mode families (cosine and tanh-shaped) give different $\lambda$ values, indicating that $\lambda$ tracks the geometry of the regions where one mode loses dominance to another.
The switching heuristic yields a test the paper does not perform: waiting times between changes of the dominant Fourier mode should be approximately exponentially distributed with mean $t_\eta$, and successive increments straddling these switches should be nearly uncorrelated; single-trajectory data could confirm or refute the mechanism directly.
Extending the numerics to $H<1/2$, where fractional increments are anti-persistent, would delimit the generality of the claim; the qualitative Brownianization may survive, but the crossover structure and the dependence of $\sigma_H$ on $H$ need not match the $H>1/2$ formulas.

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.

Desk Editor's Note

read the letter

The thing to know: this is a candid paper. The authors state up front that their main result, Theorem 1, is unproved and that they themselves do not know if it is true. The numerical evidence is real but limited to second-moment diagnostics. If you read it as a conjecture plus supporting numerics, it is solid; if you expect a proof of Brownianization, you will be disappointed, but the paper does not oversell itself.

What is genuinely new: the equal-kinetic-energy normalization that lets you compare FGN-driven models across Hurst exponents and spatial scales, and the numerical observation that a spatially disordered, divergence-free field with persistent FGN components apparently restores Brownian increments. The two-regime variance curves (slope 2H then slope 1) and the decay of the oscillation Delta are consistent with the claim, and the control case is checked against exact formulas, which gives confidence in the numerics. The authors also clearly flag that the switching mechanism in Appendix C is heuristic, not derived, and that the prefactor lambda in the diffusion formula (20) is fitted.

The soft spots are proportionate to what the paper claims. The stress-test note is right that linear variance does not imply Brownian motion: there is no direct test of increment independence or Gaussianity, so the 'memory is lost' statement rests on a necessary condition only. That is a genuine limitation, but it is not hidden—the paper calls Theorem 1 a conjecture and explicitly says the finite-eta process still 'feels' the fractional processes. The quantitative diffusion formula (20) is partly calibrated through lambda, so it is not a parameter-free prediction. No code, no error bars, and eta only down to about 7e-2 also limit how strongly one can extrapolate to the limit. Still, none of these are fatal; they are limitations the authors acknowledge.

Who is this for? Researchers in stochastic transport, turbulence parametrization, and fractional noise modeling. They will find the normalization useful and the conjectured Brownianization worth a closer look, possibly as a challenge for a rigorous homogenization proof. It deserves a serious referee: the question is important, the numerics are clean enough to check, and the paper is honest about what it does not establish. I would send it to review, with the expectation that a referee asks for a direct increment-autocorrelation diagnostic and a clear statement that the Brownian claim is numerical and conjectural.

Referee Report

3 major / 5 minor

Summary. The paper studies passive tracer dispersion in a two-dimensional divergence-free velocity field composed of many Fourier modes driven by independent fractional Gaussian noises with Hurst index H >= 1/2. It introduces a normalization constant C(eta, tau, H) that places models with different space-time structures on a common footing by matching average kinetic energy, then approximates the fractional Ornstein-Uhlenbeck velocity by a surrogate model. In a control case with constant modes the numerics are validated against exact formulas for the variance and for P(|X_t| < R). In a test case with many small-scale modes, the simulations show a crossover in the variance from slope 2H at short times to slope 1 at long times, together with a decay of the oscillation of E|X_t|^2/t as eta decreases. The paper then conjectures Theorem 1, stating that with tau = tau_eta of order eta^{(1-2H)/(1-H)} the process X_t^0 converges in law to a two-dimensional Brownian motion as eta -> 0. A heuristic switching argument in Appendix C leads to formula (20) for the diffusion coefficient sigma_H, whose prefactor lambda is fitted to the numerics.

Significance. If the Brownianization claim is correct, the paper identifies a genuinely interesting phenomenon: a spatially complex, divergence-free velocity field driven by persistent fractional noises can restore approximate independence of increments in tracer motion, leaving only a trace of H in the diffusion coefficient. This goes beyond homogenization results in which fractional behavior survives in the limit, and it is relevant for stochastic parametrizations of turbulent transport. The model normalization and the exact control-case checks are useful contributions, and the two-regime variance curves are clearly presented. The main limitation is that the central claim is supported only by second-moment diagnostics and an explicitly unproved theorem; the significance would be much higher if the paper supplied direct evidence of convergence to Brownian motion.

major comments (3)

[Sec. 2.3, Figs. 2 and 4] The central claim that X_t^0 'behaves like a Brownian Motion' and that 'the memory related to H is lost' is supported only by the slope of E|X_t^0|^2 and by the decay of Delta(t0,t1,eta,tau,H). Both are second-moment diagnostics. Linear growth of the second moment is necessary but not sufficient for Brownian motion; it does not test independence of increments or Gaussianity. Since the paper itself notes that for finite eta a residual memory must remain and the asymptotic statement is Theorem 1, the numerical evidence should include direct tests of the limiting law: for example, autocorrelation of increments at fixed lags tending to zero as eta -> 0, or convergence of the finite-dimensional distributions of the rescaled process to a Gaussian. Without such evidence the 'lost memory' claim is not established.
[Sec. 2.3, Theorem 1] Theorem 1 is labeled a theorem but is explicitly unproved; the text states 'we do not know whether the previous theorem is true or not' and offers only partial numerical verification down to eta approximately 7e-2. If this statement remains unproved it should be called a conjecture or an open problem, not a theorem, in the abstract and conclusions. More importantly, the numerical evidence in Fig. 4 and the two-slope variance in Fig. 2 do not establish convergence in law of the process; they only suggest that E|X|^2/t has a plateau. The claim of convergence to a two-dimensional Brownian motion requires either a proof or a much stronger numerical convergence test, such as distribution and covariance diagnostics at finite eta with extrapolation to the limit.
[Appendix C, Eq. (20) and Sec. 3] Formula (20) for sigma_H is presented as a theoretical prediction, but it depends on an unknown constant lambda that is fitted to the numerical data: the text reports lambda approximately 0.47 for the test case and approximately 0.41 for the generalized g(r)=tanh(Mr) case, and acknowledges that lambda is not universal. Because lambda is not derived and is model-dependent, the agreement in Fig. 3 validates only the scaling form in eta and tau, not the switching mechanism on which the derivation rests. To make Eq. (20) a predictive result, the paper should either derive lambda from the model or from a more detailed analysis of the switching process, or state clearly that sigma_H is determined up to an empirical constant. In addition, the heuristic assumption that a tracer is influenced mostly by one Fourier mode for a time of order t_eta and then switches to an independent mode is not derived from Eq. (13); it should be tested directly, for example by measuring the decorrelation time of the effective driving process and comparing it with t_eta.

minor comments (5)

[Abstract and Sec. 3] The abstract contains the typo 'the discover' and should read 'the discovery'; in Sec. 3 there is a typo 'Fig. 2)' in the text discussing the vertical lines in Fig. 2.
[Sec. 2.3, Remark 2] Remark 2 refers to 'formula (15)' but the surrounding discussion and numbering suggest it should refer either to Eq. (20) or to Claim 3; please renumber or cross-reference precisely to avoid ambiguity.
[Sec. 3] The sentence 'as suggested by Remark 6 of Appendix C' appears to reference the wrong remark; Appendix C contains Remark 7, not Remark 6, and that is the one relating the diffusion coefficient to the Kubo number.
[Sec. 2.2] In the paragraph on product rules, the citation '[Nualart]' is incomplete; it should be a proper reference to [2] or a different entry in the bibliography.
[Appendix C] The derivation of the diffusion constant mixes several rough approximations: the replacement of the displacement over [t,t+t_eta] by a sum of fractional increments, the averaging over modes, and the later identification of the displacement size with lambda eta. Please clarify which of these steps are meant to be asymptotic as eta -> 0 and which are purely heuristic, since the final formula inherits the uncertainty from all of them.

Circularity Check

2 steps flagged · score 6.0 of 10

The central Brownianization claim is not circular, but the quantitative diffusion formula (20) is fitted, not predicted: its agreement with the numerics is partly enforced by the least-squares constant λ.

fitted input called prediction [Section 3, discussion of Fig. 3 and Eq. (20)]
"In Fig. 3 σH computed from numerical simulation is shown as function of η (dots) and compared against expression (20) (solid line). A good agreement of the theoretical prediction with the numerical results is found. The free parameter λ computed using a least-squares method is approximately 0.47. The latter is, however, not a universal constant but rather it has been found dependent on σk."

Expression (20) contains the undetermined coefficient λ, and λ is obtained by a least-squares fit to the very σH values displayed in Fig. 3. Therefore the 'good agreement' between the solid line and the dots is not an independent confirmation of the diffusion-constant prediction: the amplitude has been chosen to match those data. What remains nontrivial is the η-dependence of Eq. (20), but the claimed validation of the formula as a quantitative prediction is reduced by construction to a one-parameter fit.
fitted input called prediction [Appendix C, final paragraph and Section 4 Conclusions]
"We have not found an argument to predict the coefficient λ, but we can show numerically that there exists a value providing a good fit between this formula and numerical experiments. ... A good agreement was found up to an arbitrary constant λ."

The paper itself concedes that λ is not predicted by the derivation. The diffusion approximation in Appendix C depends on an unverified switching-time ansatz, and λ absorbs the error of that ansatz. Calling the resulting agreement 'verified against numerical predictions' is therefore circular: the only quantitative content of Eq. (20) is the functional form, while the coefficient is chosen post hoc to fit the same simulation data that are offered as validation.

full rationale

The central discovery — that after a transient the test-case tracer with H > 1/2 behaves like a Brownian motion and that the FGN memory is lost — is supported by direct simulation of the variance slope (Fig. 2) and the oscillation Δ (Fig. 4), not by the fitted constant λ. That part of the claim is not circular. Theorem 1 is explicitly presented as an unproved conjecture ('We do not know whether the previous theorem is true'), so there is no pretense that it is derived from Eq. (13). The model normalization and the control-case validation are also self-contained. The circularity is confined to the quantitative diffusion coefficient: Eq. (20) is derived from a heuristic switching mechanism, contains an unspecified coefficient λ, and λ is then fitted to the numerical σH data before the same data are used to display 'a good agreement of the theoretical prediction.' This is a textbook fitted-input-called-prediction pattern. Because λ is openly described as non-universal and fitted, the central Brownianization claim does not reduce to the fit, but the paper's quantitative prediction of the diffusion constant does reduce to it by construction. That warrants a partial-circularity score of 6, not higher. The paper also provides only second-moment diagnostics for Brownian behavior and never directly tests increment independence or Gaussianity; that is an evidentiary weakness, but not itself circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on four unproved or partially supported premises: the rough-path integration framework, the surrogate replacement of the OU model by FGN, the mode-switching mechanism, and the finite-eta to limit extrapolation. The only fitted free parameter is lambda. No new physical entities are introduced.

free parameters (1)

lambda (the unknown prefactor in sigma_H) = about 0.47 for the cosine model, about 0.41 for tanh(Mr) with M=10
Appears in the heuristic diffusion coefficient formula (20); not predicted by theory and fitted by least-squares to the same numerical data used to validate the formula. The paper states it is not universal.

assumptions (4)

standard math The Lagrangian formulation (13) is the correct interpretation of transport equation (3), and for H >= 1/2 the stochastic integrals are Young or Stratonovich integrals.
This is standard for rough paths and Young integration with H > 1/2 and Stratonovich for H = 1/2; the paper cites [2], [3], [4] for the rigorous framework.
domain assumption The FGN velocity field (4) is a faithful surrogate for the OU-regularized field (5); the approximation in (7) is accurate in the regime of interest.
Lemma 6 proves an L^2 bound O(tau^{2-2H}) for the integrated difference, but the paper does not check this bound is negligible at the parameters used, especially under the scaling tau_eta to infinity of Theorem 1. The models are compared by assuming this replacement.
ad hoc to paper A tracer is influenced mostly by one Fourier mode for a duration of order t_eta, then switches to another independent mode, so successive displacements are approximately independent.
This switching hypothesis is introduced in Appendix C to build the random-walk argument leading to (20); it is not proven and the fitted constant lambda absorbs its uncertainty.
domain assumption The numerically observed decrease of Delta(eta) for eta down to about 7e-2 continues to the limit eta to 0 stated in Theorem 1.
The paper declares the eta to 0 limit computationally unfeasible and tests only finite eta; extrapolating the trend is necessary for the conjectured convergence.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Diffusion properties of small-scale fractional transport models." pith.science (2026). https://pith.science/paper/CIOFI6KJ

@misc{pith2026250608068,
  author       = {Pith},
  title        = {Pith review of: Diffusion properties of small-scale fractional transport models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CIOFI6KJ}},
  note         = {Machine review of arXiv:2506.08068}
}

read the original abstract

Stochastic transport due to a velocity field modeled by the superposition of small-scale divergence free vector fields activated by Fractional Gaussian Noises (FGN) is numerically investigated. We present two non-trivial contributions: the first one is the definition of a model where different space-time structures can be compared on the same ground: this is achieved by imposing the same average kinetic energy to a standard Ornstein-Uhlenbeck approximation, then taking the limit to the idealized white noise structure. The second contribution, based on the previous one, is the discover that a mixing spatial structure with persistent FGN in the Fourier components induces a classical Brownian diffusion of passive particles, with suitable diffusion coefficient; namely, the memory of FGN is lost in the space complexity of the velocity field.

Figures

Figures reproduced from arXiv: 2506.08068 by the authors.

**Figure 1.** Probability P[|x(t) − x(0)| < R] (left panel) and variance E[|x(t) − x(0)| 2 ] (right panel) for the control case as a function of time. Numerical values are represented by the solid lines while the exact formulae are represented by the dashed lines. Having validated our numerical code, we move on to simulate non-trivial vector fields σk. In particular, we consider the vector field defined by (10), which represents … view at source ↗

**Figure 2.** Variance E[|x(t) − x(0)| 2 ] for η = π/20 (left panel), η = π/100 (right panel) and η = π/200 (bottom panel) as a function of time. The dashed lines represents the slopes 2H and 1. The dash-dotted vertical line represents t = t ∗ . 2.3. In the plots of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Standard deviation σH as a function of η computed numerically (dots) and analytically (solid line) using (20). parameter M. By repeating the above procedure we find λ ≈ 0.41 for M = 10. We thus conclude that there exists indeed a constant λ, but it is not universal. A concise way to state our results is formulated by Claim (15). While proving this statement is challenging, we here limit ourselves to simulate the osc… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Oscillation ∆(t0, t1, η, τ, H) as a function of η computed numerically. ported in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Particle positions at t = 0 (top-left panel), t = t ∗/10 (top-right panel), t = t ∗ (mid-left panel), t = 4t ∗ (mid-right panel), t = 10t ∗ (bottomleft panel) and t = 25t ∗ (bottom-right panel). The circle r = η/8 is depicted by solid line in all figures as point of r…

**Figure 6.** Figure 6: Particles variance VARN (t) as a function of time for H = 0.7 (solid line) and H = 0.5 (dash-dotted line). Slope 1 is represented by the inclined dashed line while t = t ∗ is represented by the vertical dashed line. A Model scaling Consider the velocity field uτ (x, t)…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

25 extracted references · 20 canonical work pages

[1]

Fractional brown ian mo- tions, fractional noises and applications

Benoit B Mandelbrot and John W Van Ness. Fractional brown ian mo- tions, fractional noises and applications. SIAM review , 10(4):422–437, 1968

work page 1968
[2]

The Malliavin calculus and related topics , volume 1995

David Nualart. The Malliavin calculus and related topics , volume 1995. Springer, 2006

1995
[3]

Skorohod calculus and paley-wiener in tegrals for covariance singular processes (2010)

I Kruk and F Russo. Skorohod calculus and paley-wiener in tegrals for covariance singular processes (2010). arxiv

work page 2010
[4]

Reduced dissipati on eﬀect in stochastic transport by gaussian noise with regularity gre ater than 1/2

Franco Flandoli and Francesco Russo. Reduced dissipati on eﬀect in stochastic transport by gaussian noise with regularity gre ater than 1/2. arXiv preprint arXiv:2305.19293 , 2023

arXiv 2023
[5]

A linear stochastic model of tur bulent cascades and fractional ﬁelds

Gabriel B Apolin´ ario, Geoﬀrey Beck, Laurent Chevillard , Isabelle Gal- lagher, and Ricardo Grande. A linear stochastic model of tur bulent cascades and fractional ﬁelds. arXiv preprint arXiv:2301.00780 , 2023

arXiv 2023
[6]

Dynamical fractional and multifractal ﬁelds

Gabriel B Apolin´ ario, Laurent Chevillard, and Jean-Ch ristophe Mour- rat. Dynamical fractional and multifractal ﬁelds. Journal of Statistical Physics, 186(1):15, 2022

work page 2022
[7]

Mu l- tifractal fractional ornstein-uhlenbeck processes

Laurent Chevillard, Marc Lagoin, and Stephane G Roux. Mu l- tifractal fractional ornstein-uhlenbeck processes. arXiv preprint arXiv:2011.09503, 2020

arXiv 2011
[8]

Fractional bro wnian mo- tions and enhanced diﬀusion in a unidirectional wave-like tu rbulence

Albert Fannjiang and Tomasz Komorowski. Fractional bro wnian mo- tions and enhanced diﬀusion in a unidirectional wave-like tu rbulence. Journal of Statistical Physics , 100:1071–1095, 2000. 21

work page 2000

Show all 25 references

[9]

Fractional bro wnian mo- tions in a limit of turbulent transport

Albert Fannjiang and Tomasz Komorowski. Fractional bro wnian mo- tions in a limit of turbulent transport. Annals of Applied Probability , pages 1100–1120, 2000

2000
[10]

Frozen path ap proximation for turbulent diﬀusion and fractional brownian motion in ran dom ﬂows

Albert Fannjiang and Tomasz Komorowski. Frozen path ap proximation for turbulent diﬀusion and fractional brownian motion in ran dom ﬂows. SIAM Journal on Applied Mathematics , 63(6):2042–2062, 2003

2003
[11]

H omoge- nization driven by a fractional brownian motion: the shear l ayer case

Tomasz Komorowski, Alexei Novikov, and Lenya Ryzhik. H omoge- nization driven by a fractional brownian motion: the shear l ayer case. Multiscale Modeling & Simulation , 12(2):440–457, 2014

2014
[12]

Pa ssive advection of fractional brownian motion by random layered ﬂows

Alessio Squarcini, Enzo Marinari, and Gleb Oshanin. Pa ssive advection of fractional brownian motion by random layered ﬂows. New Journal of Physics , 22(5):053052, 2020

2020
[13]

On the convergence of stochastic transp ort equations to a deterministic parabolic one

Lucio Galeati. On the convergence of stochastic transp ort equations to a deterministic parabolic one. Stochastics and Partial Diﬀerential Equations: Analysis and Computations , 8(4):833–868, 2020

2020
[14]

Stochastic partial diﬀerential equations in ﬂuid mechanics , volume 2330

Franco Flandoli, Eliseo Luongo, et al. Stochastic partial diﬀerential equations in ﬂuid mechanics , volume 2330. Springer, 2023

2023
[15]

Enhanced dissipation for stochastic navier –stokes equations with transport noise

Dejun Luo. Enhanced dissipation for stochastic navier –stokes equations with transport noise. Journal of Dynamics and Diﬀerential Equations , 37(1):859–894, 2025

2025
[16]

Stochastic transport in upper ocean dynamic s

Bertrand Chapron, Dan Crisan, Darryl Holm, Etienne M´ e min, and Anna Radomska. Stochastic transport in upper ocean dynamic s. stuod 2021 workshop, london, uk, september 20–23. 2023

2021
[17]

Solution properties of the incompressible euler s ystem with rough path advection

Dan Crisan, Darryl D Holm, James-Michael Leahy, and Tor stein Nilssen. Solution properties of the incompressible euler s ystem with rough path advection. Journal of Functional Analysis , 283(9):109632, 2022

2022
[18]

Variational principles for ﬂuid dynamics on rough paths

Dan Crisan, Darryl D Holm, James-Michael Leahy, and Tor stein Nilssen. Variational principles for ﬂuid dynamics on rough paths. arXiv preprint arXiv:2004.07829, 2020

2004 arXiv
[19]

Data-driven stochastic lie transport modeling of the 2d eul er equations

Sagy R Ephrati, Paolo Cifani, Erwin Luesink, and Bernar d J Geurts. Data-driven stochastic lie transport modeling of the 2d eul er equations. Journal of Advances in Modeling Earth Systems , 15(1):e2022MS003268, 2023

2023
[20]

Variational principles for stochastic ﬂ uid dynamics

Darryl D Holm. Variational principles for stochastic ﬂ uid dynamics. Proceedings of the Royal Society A: Mathematical, Physical and Engi- neering Sciences, 471(2176):20140963, 2015. 22

2015
[21]

Stochastic me soscale circulation dynamics in the thermal ocean

Darryl D Holm, Erwin Luesink, and Wei Pan. Stochastic me soscale circulation dynamics in the thermal ocean. Physics of Fluids , 33(4), 2021

2021
[22]

Fluid ﬂow dynamics under location unce rtainty

Etienne M´ emin. Fluid ﬂow dynamics under location unce rtainty. Geo- physical & Astrophysical Fluid Dynamics , 108(2):119–146, 2014

2014
[23]

Geophys- ical ﬂows under location uncertainty, part i random transpo rt and gen- eral models

Valentin Resseguier, Etienne M´ emin, and Bertrand Chapron. Geophys- ical ﬂows under location uncertainty, part i random transpo rt and gen- eral models. Geophysical & Astrophysical Fluid Dynamics , 111(3):149– 176, 2017

2017
[24]

Simpliﬁed models for t urbu- lent diﬀusion: theory, numerical modelling, and physical ph enomena

Andrew J Majda and Peter R Kramer. Simpliﬁed models for t urbu- lent diﬀusion: theory, numerical modelling, and physical ph enomena. Physics reports, 314(4-5):237–574, 1999

1999
[25]

Gyrokinetic simulations of turbulent transport

Xavier Garbet, Yasuhiro Idomura, Laurent Villard, and TH Watan- abe. Gyrokinetic simulations of turbulent transport. Nuclear Fusion, 50(4):043002, 2010. 23

2010

Pith tools

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

[1] [1]

Fractional brown ian mo- tions, fractional noises and applications

Benoit B Mandelbrot and John W Van Ness. Fractional brown ian mo- tions, fractional noises and applications. SIAM review , 10(4):422–437, 1968

work page 1968

[2] [2]

The Malliavin calculus and related topics , volume 1995

David Nualart. The Malliavin calculus and related topics , volume 1995. Springer, 2006

1995

[3] [3]

Skorohod calculus and paley-wiener in tegrals for covariance singular processes (2010)

I Kruk and F Russo. Skorohod calculus and paley-wiener in tegrals for covariance singular processes (2010). arxiv

work page 2010

[4] [4]

Reduced dissipati on eﬀect in stochastic transport by gaussian noise with regularity gre ater than 1/2

Franco Flandoli and Francesco Russo. Reduced dissipati on eﬀect in stochastic transport by gaussian noise with regularity gre ater than 1/2. arXiv preprint arXiv:2305.19293 , 2023

arXiv 2023

[5] [5]

A linear stochastic model of tur bulent cascades and fractional ﬁelds

Gabriel B Apolin´ ario, Geoﬀrey Beck, Laurent Chevillard , Isabelle Gal- lagher, and Ricardo Grande. A linear stochastic model of tur bulent cascades and fractional ﬁelds. arXiv preprint arXiv:2301.00780 , 2023

arXiv 2023

[6] [6]

Dynamical fractional and multifractal ﬁelds

Gabriel B Apolin´ ario, Laurent Chevillard, and Jean-Ch ristophe Mour- rat. Dynamical fractional and multifractal ﬁelds. Journal of Statistical Physics, 186(1):15, 2022

work page 2022

[7] [7]

Mu l- tifractal fractional ornstein-uhlenbeck processes

Laurent Chevillard, Marc Lagoin, and Stephane G Roux. Mu l- tifractal fractional ornstein-uhlenbeck processes. arXiv preprint arXiv:2011.09503, 2020

arXiv 2011

[8] [8]

Fractional bro wnian mo- tions and enhanced diﬀusion in a unidirectional wave-like tu rbulence

Albert Fannjiang and Tomasz Komorowski. Fractional bro wnian mo- tions and enhanced diﬀusion in a unidirectional wave-like tu rbulence. Journal of Statistical Physics , 100:1071–1095, 2000. 21

work page 2000