Limit Theorems for the Dynamical Foundation of the Fractional Brownian Motion and Related Models of Anomalous Diffusion with Random Diffusion Coefficient and Time-Dependent Random Hurst parameter

Christian Bender; Gianni Pagnini; Mirko D'Ovidio; Yana A. Butko

REVIEW 3 major objections 4 minor 1 cited by

Limit Theorems for the Dynamical Foundation of the Fractional Brownian Motion and Related Models of Anomalous Diffusion with Random Diffusion Coefficient and Time-Dependent Random Hurst parameter

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A test particle kicked by many unequal-mass Brownian particles converges, as N grows, to a Gaussian process whose covariance is set by the mass distribution; for a power-law mass law the limit is fractional Brownian motion with Hurst…

desk verdict Rigorous limit theorem deriving fBm and superstatistical fBm with random Hurst parameter from a heterogeneous-mass Langevin system; the conservation-of-momentum claim is overstated and should be fixed, but the mathematics holds up. read the letter →

arxiv 2411.18775 v2 pith:OUNZDI5M submitted 2024-11-27 math.PR math-phmath.MP

Christian Bender , Yana A. Butko , Mirko D'Ovidio , Gianni Pagnini This is my paper · ORCID

classification math.PRmath-phmath.MP MSC 60F1760G1560G2260J60

keywords fractionalBrownianmotionsuperstatisticalgeneralizedgreyrandomlyscaledGaussianprocessesrandomHurstparametertime-dependentanomalousdiffusionensembleheterogeneity

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

The paper tries to show that fractional Brownian motion and its random-coefficient, random-Hurst neighbors are limiting behaviors of a concrete mechanical system, not just fitting models. A test particle of mass $M$ is coupled to $N$ underdamped Langevin particles whose masses are random and whose couplings decay as prescribed powers of $N$. The authors prove that as $N\to\infty$ the test-particle position converges to a centered Gaussian process with covariance $D(v(t)+v(s)-v(|t-s|))$, where $v$ is read off from the limiting mass distribution. Choosing the surround masses according to a stable subordinator's L\'evy measure yields fBm with $H\in(1/2,1)$, while mixtures of power laws yield a sum of independent fBms whose effective anomalous exponent changes with time. Randomizing the test-particle coupling and the mass law through a random environment parameter $H$ then produces superstatistical fBm and models with time-dependent random Hurst parameter, together with their generalized Fokker--Planck equations.

What carries the argument

The load-bearing object is the covariance identity $\operatorname{Cov}(Z_t,Z_s)=D(v(t)+v(s)-v(|t-s|))$ with $v(t)=\int_0^t \dot v(\tau)d\tau$, where $\dot v$ is the pointwise limit of $e_N(t)/m^*_N$ and $e_N(t)=\int(1-e^{-\gamma y t})y^{-2}\mu_N(dy)$. This $e_N$ is an integrated Laplace transform of the mass law, so the family of limits is organized by Bernstein functions when the masses come from subordinator L\'evy measures. The proof mechanism is conditional Gaussianity: given the random masses, the surround velocities are Ornstein--Uhlenbeck processes, and the paper proves $\sigma(M)$-mixing convergence, with the exact exponent balance $2(a-b)-\delta=1$ making the error terms from the cross-interaction and coupling vanish as $N\to\infty$.

What would settle it

Simulate system (6) with couplings $\alpha_{k,N}\simeq C_\alpha N^{-a}$, $\beta_{k,N}\simeq C_\beta N^{-b}$ and masses drawn from the $(2H-1)$-stable subordinator construction; the theorem predicts convergence to the stated fBm covariance when $2(a-b)-\delta=1$ and $b>d$, so observing that same covariance when the balance is broken would refute the claim, while divergence or a different covariance under the broken balance would support it.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 2.1: under scaling assumptions, the position process $X^N$ solving the Langevin system (6) converges in finite-dimensional distributions (and, with an extra regularity condition, in $C[0,t_0]$) to a centered Gaussian process $Z$ with covariance $\operatorname{Cov}(Z_t,Z_s)=\left(2\sigma C_\beta^2 C_\delta/(\gamma^2 C_\alpha^2)\right)^{1/2}(v(t)+v(s)-v(|t-s|))$, where $v(t)=\int_0^t \dot v(\tau)d\tau$ and $\dot v$ is the monotone limit of $e_N(t)/m^*_N$, with $e_N(t)=\int_{(0,\infty)}(1-e^{-\gamma y t})y^{-2}\mu_N(dy)$. The variance function $v$ is the fingerprint of the surround-mass distribution: when $\mu_N$ is built from the L\'evy measure of the $(2H-1)$-stable subordinator, $v(t)\propto t^{2H}$, so the limit is fBm with Hurst parameter $H\in(1/2,1)$. Mixtures of power laws give a sum of independent fBms and a time-dependent anomalous exponent; tempered or exponential mass laws interpolate between ballistic and superdiffusive or classical regimes; a deterministic mass choice recovers a Wiener process. The paper then randomizes the coupling constant $A$ and the mass distribution through a random $H$, obtaining conditionally Gaussian limits $\sqrt{A}\,G^{(H)}$ that include a randomly scaled (superstatistical) fBm with random Hurst parameter.

Load-bearing premise

The proof requires the exact exponent balance $2(a-b)-\delta=1$ and the inequality $b>d$; these are tuning conditions imposed on the coupling and mass scalings, and if the scalings deviate, the error terms do not vanish and the claimed limit need not hold.

Editorial extensions

If this is right

Superstatistical fBm, previously proposed as a phenomenological model, is obtained as the $N\to\infty$ limit of a momentum- and energy-conserving Langevin system, giving it a dynamical foundation.
The same theorem produces fBm-like Gaussian limits whose anomalous exponent changes with time, such as sums of independent fBms with different Hurst parameters.
With random coupling $A$ and random environment variable $H$, the limit $\sqrt{A}\,G^{(H)}$ covers both random diffusion coefficient and random Hurst parameter in one conditionally Gaussian process.
The associated Kolmogorov--Fokker--Planck equations are generalized evolution equations with pseudo-differential operators, so users of these anomalous-diffusion models also have their generators.
Because the limiting covariance is determined by the mass distribution, the listed choices of mass laws give concrete predictions: fBm, mixtures of fBms, ballistic-to-superdiffusive and ballistic-to-classical crossovers, and ordinary diffusion.

Reading between the lines

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

A testable extension the authors leave implicit: the measured variance function $v(t)$ of a tracer should map back to the effective crowd-mass distribution through $e_N(t)/m^*_N$, so single-particle-tracking data on $v(t)$ constrain the environmental heterogeneity rather than leaving Hurst parameter free.
The tuning condition $2(a-b)-\delta=1$ suggests anomalous scaling may be confined to an intermediate window in particle number and coupling strength; finite-$N$ simulations of system (6) away from that window could show crossovers that the infinite-$N$ theorem does not describe.
The Bernstein-function organization of the examples indicates the construction generalizes: any mass law whose $e_N(t)/m^*_N$ converges to a nondecreasing Bernstein-like function should generate a valid Gaussian limit, potentially yielding new anomalous-diffusion processes by choosing $\Phi$ at will.
In the random-$H$ construction, each value $h$ can be read as a local environment patch; spatial segmentation of trajectories should then reproduce the conditional covariance $v_h$ within patches, a prediction accessible to experiments with spatially varying crowding.

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

Dear colleague,

The headline: this is a rigorous limit theorem, the first of its kind, turning an underdamped Langevin system with heterogeneous masses into fBm and, with extra randomization, superstatistical fBm with random Hurst parameter. The proof is clean and self-contained; explicit error estimates, mixing convergence, and the examples all check out. The stable-subordinator case yields H in (1/2,1); mixtures give time-dependent anomalous exponents; the random-Cα and random-H extension is a natural consequence of Theorem 2.1. The paper deserves a serious referee.

The soft spot is the advertised physical interpretation. The abstract and Section 2 claim conservation of momentum and energy. For system (6), total momentum P_t = M V_t^N + Σ m_k U_t^{k,N} obeys dP_t = Σ(β_k − γ m_k²) U^k dt + Σ m_k sqrt(2σ) dW^k; the α-coupling cancels, but the thermostat damping and noise do not. So the system does not conserve momentum. This does not affect the limit theorem, whose proof uses only the linear SDE structure, but it does undercut the 'dynamical foundation' language. The authors should either say the coupling terms are constructed to conserve momentum in the interaction (action–reaction), or present the system as a Langevin thermostat around a conservative coupling. As written, the claim is too strong.

Minor issues: Assumption 2.6 imposes the exact balance 2(a−b)−δ=1 and b>d. That's a tuning condition, typical for this kind of scaling limit, but the paper would be clearer if it said so explicitly rather than presenting it as if the mechanics forced it. In Remark 3.2, the KFP derivation needs a bit more regularity on v_H and the Laplace transform to justify differentiating under the expectation and dividing by it. Also, in Example 2.2(iii), the gamma subordinator's Laplace exponent is Φ(λ)=log(1+λ); the paper writes log(1−λ), which is a typo (the second listed pair is fine).

Bottom line: the reviewer's conditional verdict is right. The mathematics holds; the interpretation needs reworking. I'd send it to review.

Best,

Referee Report

3 major / 4 minor

Summary. The paper studies a test particle coupled to N underdamped Langevin bath particles with random, heterogeneous masses and with coupling constants that scale as $\alpha_{k,N}\simeq C_\alpha N^{-a}$, $\beta_{k,N}\simeq C_\beta N^{-b}$. The main result (Theorem 2.1) states that, as $N\to\infty$, the test-particle position converges to a centered Gaussian process with stationary increments and covariance $D_0(v(t)+v(s)-v(|t-s|))$, where $v$ is determined by the limiting mass distribution; for a stable-subordinator mass distribution this gives fractional Brownian motion with $H\in(1/2,1)$. Section 3 adds a random coupling coefficient $A$ and a random Hurst-type parameter $H$, yielding conditionally Gaussian superstatistical models, and Remark 3.2 derives associated Kolmogorov--Fokker--Planck equations. The proof is self-contained and proceeds through three explicit approximation steps, using mixing convergence and detailed $L^2$ error estimates.

Significance. The mathematical core is a genuine limit theorem: the proof is self-contained, all lemmas carry explicit error rates, the mass distribution is an input rather than a fitted target, and no part of the argument is circular. This gives a rigorous route from a particle system to fractional Brownian motion and to a broad family of related Gaussian and conditionally Gaussian processes, which is of real value for the anomalous-diffusion literature. However, two load-bearing presentation points need attention: the advertised conservation of momentum is false for system (6), and the covariance constant in the main theorem statement is inconsistent with the proof and with the examples. These issues are correctable but currently undermine the physical interpretation and the exact statement of the limit.

major comments (3)

[Theorem 2.1, Lemma 4.4, Example 3.1] The limiting covariance prefactor in Theorem 2.1 and Lemma 4.4 is inconsistent with the proof. From equations (34) and (35), one obtains $E[\operatorname{Cov}(\tilde Z^N_t,\tilde Z^N_s\mid M)]\to (\sigma/\gamma^2)(C_\beta^2C_\delta/C_\alpha^2)(v(t)+v(s)-v(|t-s|))$, because the factor $1/2$ in (34) combines with the factor $2\sigma/\gamma^2$ in (35) and because $2(a-b)-\delta=1$ kills the $N$ dependence. The theorem and Lemma 4.4 instead display $\bigl(2\sigma C_\beta^2C_\delta/(\gamma^2C_\alpha^2)\bigr)^{1/2}$ as the prefactor. Remark 2.1's constant $D=\sigma C_\beta^2C_\delta/(\gamma^2C_\alpha^2)$ agrees with the derivation from (34)--(35), while the theorem statement and the formulas in Example 3.1(i) do not. This must be reconciled throughout the paper, including the displayed special cases.
[Abstract, Section 2] The paper claims that conservation of momentum is met by system (6). This is not the case. For $P_t=M V^N_t+\sum_{k=1}^N m_{k,N}U^{k,N}_t$, direct differentiation gives $dP_t=\sum_{k=1}^N(\beta_{k,N}-\gamma m_{k,N}^2)U^{k,N}_t\,dt+\sum_{k=1}^N m_{k,N}\sqrt{2\sigma}\,dW^k_t$: the $\alpha$ terms cancel, while the $\beta$ coupling and the independent thermal noise remain. Hence total momentum is not conserved. The limit theorem does not use momentum conservation, so the mathematical result survives, but the advertised 'dynamical foundation' overstates what system (6) delivers. Please either restrict the conservation statement to the coupling forces and the fluctuation--dissipation relation, or provide a genuinely momentum-conserving Hamiltonian or Newtonian system with the same scaling limit.
[Remark 3.2] The derivation of the generalized Kolmogorov--Fokker--Planck equation differentiates $\mathbb E[e^{-ADv_H(s)p^2/2}]$ with respect to $s$ and then applies the chain rule and Fubini's theorem. For the general $v_h$ arising from Bernstein functions, $v_h$ is only continuous and nondecreasing, not necessarily differentiable. The argument therefore requires additional regularity assumptions on $v_h$ and dominated-convergence conditions for the interchange of expectation and differentiation. Please state these hypotheses explicitly, or present the result as a mild/integral evolution equation rather than a differential one.

minor comments (4)

[Example 2.2] For $\nu(dy)=e^{-y}y^{-1}dy$ the Laplace exponent is $\Phi(\lambda)=\log(1+\lambda)$, not $\log(1-\lambda)$; the displayed formula is undefined for $\lambda>1$ and is not a Bernstein function.
[Section 3] The symbol $H$ is used both for the random vector and for its state space $\mathcal H$; this is confusing in Example 3.1 and should be resolved by a distinct notation for the state space.
[Assumption 2.6] The exact exponent balance $2(a-b)-\delta=1$ and the inequality $b>d$ are imposed as scaling assumptions, and the proof shows that they are precisely what makes the error estimates vanish. It would be helpful to state explicitly whether they are also necessary and to comment on what happens if they fail by a small amount, since the advertised physical foundation currently rests on this tuning.
[Remark 3.2] The text contains the typo 'Schwarz space'; it should be 'Schwartz space'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the limit theorem computes its stated covariance from the prescribed mass distribution rather than assuming the target process.

full rationale

The derivation is self-contained. Theorem 2.1 takes the mass distribution mu_N and its scaling limit v-dot (Assumption 2.3) as inputs, and Lemma 4.4 computes the limiting covariance (2 sigma C_beta^2 C_delta/(gamma^2 C_alpha^2))^{1/2}(v(t)+v(s)-v(|t-s|)) from the conditional covariance of the Ornstein-Uhlenbeck processes; the covariance is not imposed as a target. The fractional Brownian motion case is an explicit calculation: substituting the (2H-1)-stable subordinator Levy measure into the subordinator construction gives Phi(lambda)=Gamma(2-2H)/(2H-1) lambda^{2H-1} and v(t)=C t^{2H} (Example 2.1(i)), so the fBm emerges from the hypothesis rather than being fitted. Assumption 2.6 is a stated scaling-condition, not a parameter fitted to the conclusion. Citations to the authors' prior work ([14], [8], [43], [44]) are contextual or illustrative; the proof of Theorem 2.1 relies on Lemmas 4.1-4.10 proved inside the paper, together with standard stable-convergence results [22]. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work to force the choice of process. The only notable flaw is the physical claim in the abstract and introduction that conservation of momentum is met: for system (6), dP_t = sum_k (beta_{k,N} - gamma m_{k,N}^2) U^{k,N}_t dt + sum_k m_{k,N} sqrt(2 sigma) dW^k_t, so total momentum is not conserved. This is a correctness/physics concern, not circularity, and it does not affect the mathematical limit theorem.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper contains no data and no fitting. The scaling exponents in Assumption 2.6 and the Hurst parameter in examples are chosen inputs, not parameters adjusted to match a target output. The axioms list the structural assumptions under which the limit theorem is proved.

free parameters (5)

a
Scaling exponent of the damping coupling α_k,N ~ C_α N^{-a}; restricted by Assumption 2.6, not fitted to data.
b
Scaling exponent of the driving coupling β_k,N ~ C_β N^{-b}; restricted by Assumption 2.6.
δ
Scaling of m*_N ~ C_δ N^{-δ}; linked to a-b by 2(a-b)-δ=1 in Assumption 2.6.
d
Scaling of the lower mass cutoff m_min^N ~ N^{-d}; must satisfy b>d and other inequalities in Assumption 2.6.
H
Hurst parameter in the fBm examples, chosen as input; this is a target label rather than a fitted constant.

assumptions (6)

domain assumption Surround-particle velocities follow Ornstein-Uhlenbeck processes with friction γ m_k and noise √(2σ) dW_t (system 6).
This underdamped Langevin form, with Assumptions 2.1 and 2.5, is the physical basis of the model; the proof is built on the OU covariance in Lemma 4.1.
domain assumption Fluctuation-dissipation relation (β_0+β) k_B T = σ_0 (Eq. 4).
Used to set the noise amplitude and to express γ=σ/(k_B T).
ad hoc to paper Coupling constants scale as α_k,N ≃ C_α N^{-a}, β_k,N ≃ C_β N^{-b} (Assumption 2.2).
Chosen to make the N→∞ limit non-degenerate; no physical law forces these powers.
ad hoc to paper Exact exponent balance 2(a-b)-δ=1 and b>d (Assumption 2.6).
Tuning condition; the proof's error estimates depend on it.
domain assumption Mass distribution admits limits e_N/m*_N ↑ ˙v and ∫ y^{-4} μ_N(dy) ≲ N^{d'} (Assumption 2.3(i)).
Defines the variance profile v and ensures the covariance converges.
domain assumption Gaussian initial velocities for the surround, N(0, σ/(γ m)) (Assumption 2.5).
Gives the OU process stationary initial distribution, simplifying the covariance computation in Lemma 4.1.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Limit Theorems for the Dynamical Foundation of the Fractional Brownian Motion and Related Models of Anomalous Diffusion with Random Diffusion Coefficient and Time-Dependent Random Hurst parameter." pith.science (2026). https://pith.science/paper/OUNZDI5M

@misc{pith2026241118775,
  author       = {Pith},
  title        = {Pith review of: Limit Theorems for the Dynamical Foundation of the Fractional Brownian Motion and Related Models of Anomalous Diffusion with Random Diffusion Coefficient and Time-Dependent Random Hurst parameter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OUNZDI5M}},
  note         = {Machine review of arXiv:2411.18775}
}

abstract

Anomalous diffusion is an established phenomenon but still a theoretical challenge in non-equilibrium statistical mechanics. Physical models are built incrementally, and the most recent and most general family is based on the fractional Brownian motion (fBm) with a random diffusion coefficient (superstatistical fBm) together with a time-dependent random Hurst parameter. We provide here a dynamical foundation for such general family of models. We consider a dynamical system describing the motion of a test-particle surrounded by $N$ Brownian particles with different masses. This dynamic is governed by underdamped Langevin equations. Physical principles of conservation of momentum and energy are met. We prove that, in the limit $N\to\infty$, the test-particle diffuses in time according to a quite general (non-Markovian) Gaussian process whose covariance function is determined by the distribution of the masses of the surround-particles. In particular, with proper choices of the distribution of the masses of the surround-particles, we obtain fBm together with a number of other special cases of interest in modelling anomalous diffusion including time-dependent anomalous exponent. Furthermore, when the ensemble heterogeneity of the surround-particles embodying the environment becomes non-uniform and joins with the individual inhomogeneity of the test-particles, we show that, in the limit $N\to\infty$, the test-particle diffuses in time according to a quite general conditionally Gaussian process that can be calibrated into a fBm with random diffusion coefficient and random time-dependent Hurst parameter. We conclude our study by reporting the generalised Kolmogorov--Fokker--Planck equations associated to these highly general processes.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Discrete-space and -time analogue of a super-diffusive fractional Brownian motion
cond-mat.stat-mech 2025-06 conditional novelty 4.0 of 10

Sign-thresholding or magnitude-rounding of fractional Brownian motion increments yields lattice random walks with the same power-law correlations as super-diffusive fBm.

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages · cited by 1 Pith paper

[1]

Balcerek, K

M. Balcerek, K. Burnecki, S. Thapa, A. Wy lomanska, and A. Chechkin. Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions. Chaos, 32(9):093114, 09 2022

work page 2022
[2]

Balcerek, S

M. Balcerek, S. Thapa, K. Burnecki, H. Kantz, R. Metzler, A. Wy loma´ nska, and A. Chechkin. Multifractional Brownian motion with telegraphic, stochastically varying exponent. Phys. Rev. Lett., 134:197101, 2025

work page 2025
[3]

Barkai, Y

E. Barkai, Y. Garini, and R. Metzler. Strange kinetics of single molecules in living cells. Phys. Today, 65(8):29, 2012

work page 2012
[4]

Beghin, L

L. Beghin, L. Cristofaro, and J. Gajda. Non-Gaussian measures in infinite dimensional spaces: the Gamma-grey noise. Potential Anal., 60:1571–1593, 2023

work page 2023
[5]

Bender, M

C. Bender, M. Bormann, and Y. A. Butko. Subordination principle and Feynman–Kac for- mulae for generalized time-fractional evolution equations. Fract. Calc. Appl. Anal., 25:1818– 1836, 2022

work page 2022
[6]

Bender and Y

C. Bender and Y. A. Butko. Stochastic solutions of generalized time-fractional evolution equations. Fract. Calc. Appl. Anal. , 25:488–519, 2022

work page 2022
[7]

Benelli and M

R. Benelli and M. Weiss. From sub-to superdiffusion: Fractional Brownian motion of mem- braneless organelles in early C. elegans embryos. New J. Phys , 23:063072, 2021

work page 2021
[8]

Y. A. Butko and M. Mlinarzik. Fractional Ito calculus for randomly scaled fractional Brow- nian motion and its applications to evolution equations. Preprint, 2024

work page 2024

Show all 62 references

[9]

A. V. Chechkin, F. Seno, R. Metzler, and I. M. Sokolov. Brownian yet non-Gaussian diffusion: From superstatistics to subordination of diffusing diffusivities. Phys. Rev. X, 7:021002, 2017

2017
[10]

A. G. Cherstvy, S. Thapa, C. E. Wagner, and R. Metzler. Non-Gaussian, non-ergodic, and non-Fickian diffusion of tracers in mucin hydrogels. Soft Matter , 15(12):2526–2551, 2019

2019
[11]

Chubynsky and G

M. Chubynsky and G. Slater. Diffusing diffusivity: A model for anomalous, yet Brownian, diffusion. Phys. Rev. Lett. , 113:098302, 2014

2014
[12]

M. A. F. dos Santos and L. Menon Junior. Log-Normal superstatistics for Brownian particles in a heterogeneous environment. Physics, 2:571–586, 2020. 32

2020
[13]

M. A. F. dos Santos and L. Menon Junior. Random diffusivity models for scaled Brownian motion. Chaos Solitons Fract., 144:110634, 2021

2021
[14]

D’Ovidio, S

M. D’Ovidio, S. Vitali, V. Sposini, O. Sliusarenko, P. Paradisi, G. Castellani, and G. Pagnini. Centre-of-mass like superposition of Ornstein–Uhlenbeck processes: A pathway to non- autonomous stochastic differential equations and to fractional diffusion. Fract. Calc. Appl. Ana...

2018
[15]

Drakopoulos, Z

A. Drakopoulos, Z. Koszegi, Y. Lanoisel´ ee, H. H¨ ubner, P. Gmeiner, D. Calebiro, and M. Decker. Investigation of inactive-state kappa opioid receptor homodimerization via single- molecule microscopy using new antagonistic fluorescent probes. J. Med. Chem. , 63(7):3596– 3609, 2020

2020
[16]

F. Etoc, E. Balloul, C. Vicario, D. Normanno, A. Liße Domenik, Sittner, J. Piehler, M. Da- han, and M. Coppey. Non-specific interactions govern cytosolic diffusion of nanosized objects in mammalian cells. Nat. Mater. , 17:740–746, 2018

2018
[17]

Golding and E

I. Golding and E. C. Cox. Physical nature of bacterial cytoplasm. Phys. Rev. Lett., 96:098102, 2006

2006
[18]

P. Grassia. Dissipation, fluctuations, and conservation laws. Am. J. Phys., 69:113–119, 2001

2001
[19]

Grimes, Z

J. Grimes, Z. Koszegi, Y. Lanoisel´ ee, T. Miljus, S. O’Brien, T. Stepniewski, B. Medel– Lacruz, M. Baidya, M. Makarova, R. Mistry, J. Goulding, J. Drube, C. Hoffmann, D. Owen, A. Shukla, J. Selent, S. Hill, and D. Calebiro. Plasma membrane preassociation drives β-arrestin cou...

2023
[20]

Grothaus and F

M. Grothaus and F. Jahnert. Mittag–Leffler analysis II: Application to the fractional heat equation. J. Funct. Anal. , 270:2732–2768, 2016

2016
[21]

Grothaus, F

M. Grothaus, F. Jahnert, F. Riemann, and J. L. da Silva. Mittag–Leffler analysis I: Con- struction and characterization. J. Funct. Anal. , 268:1876–1903, 2015

1903
[22]

H¨ ausler and H

E. H¨ ausler and H. Luschgy. Stable convergence and stable limit theorems , volume 74 of Probab. Theory Stoch. Model. Cham: Springer, 2015

2015
[23]

H¨ ofling and T

F. H¨ ofling and T. Franosch. Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys., 76:046602, 2013

2013
[24]

Itto and C

Y. Itto and C. Beck. Superstatistical modelling of protein diffusion dynamics in bacteria. J. R. Soc. Interface , 18:20200927, 2021

2021
[25]

Janczura, M

J. Janczura, M. Balcerek, K. Burnecki, A. Sabri, M. Weiss, and D. Krapf. Identifying heterogeneous diffusion states in the cytoplasm by a hidden Markov model. New J. Phys. , 23(5):053018, 2021

2021
[26]

Kallenberg

O. Kallenberg. Foundations of modern probability. Probab. Appl. New York, NY: Springer, 1997. 33

1997
[27]

Klafter, S.-C

J. Klafter, S.-C. Lim, and R. Metzler, editors. Fractional Dynamics: Recent Advances. World Scientific, Singapore, 2012

2012
[28]

Klafter and I

J. Klafter and I. M. Sokolov. Anomalous diffusion spread its wings. Physics World, 18:29–32, 2005

2005
[29]

Klages, G

R. Klages, G. Radons, and I. M. Sokolov, editors. Anomalous Transport: Foundations and Applications. Wiley–VCH Verlag GmbH & Co. KGaA, Weinheim, 2008

2008
[30]

Korabel, D

N. Korabel, D. Han, A. Taloni, G. Pagnini, S. Fedotov, V. Allan, and T. A. Waigh. Local analysis of heterogeneous intracellular transport: slow and fast moving endosomes. Entropy, 23:958, 2021

2021
[31]

Korabel, A

N. Korabel, A. Taloni, G. Pagnini, V. Allan, S. Fedotov, and T. A. Waigh. Ensemble heterogeneity mimics ageing for endosomal dynamics within eukaryotic cells. Sci. Rep. , 13:8789, 2023

2023
[32]

Lanoisel´ ee, G

Y. Lanoisel´ ee, G. Pagnini, and A. Wy loma´ nska. Super-resolved anomalous diffu- sion: deciphering the joint distribution of anomalous exponent and diffusion coefficient. arXiv:2410.18133
[33]

Ma´ cka la and M

A. Ma´ cka la and M. Magdziarz. Statistical analysis of superstatistical fractional Brownian motion and applications. Phys. Rev. E , 99:012143, 2019

2019
[34]

Magdziarz, A

M. Magdziarz, A. Weron, K. Burnecki, and J. Klafter. Fractional Brownian motion versus the Continuous-Time Random Walk: A simple test for subdiffusive dynamics. Phys. Rev. Lett., 103:180602, 2009

2009
[35]

Manzo and M

C. Manzo and M. F. Garcia-Parajo. A review of progress in single particle tracking: from methods to biophysical insights. Rep. Progr. Phys., 78:124601, 2015

2015
[36]

Manzo, J

C. Manzo, J. A. Torreno–Pina, P. Massignan, G. J. Lapeyre, M. Lewenstein, and M. F. Garcia–Parajo. Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity. Phys. Rev. X , 5:011021, 2015

2015
[37]

Metzler and J

R. Metzler and J. Klafter. The restaurant at the end of the random walk: recent developments in fractional dynamics descriptions of anomalous dynamical processes. J. Phys. A: Math. Theor., 37(31):R161–R208, 2004

2004
[38]

Molina-Garc ´ ıa, T

D. Molina-Garc ´ ıa, T. Minh Pham, P. Paradisi, C. Manzo, and G. Pagnini. Fractional kinetics emerging from ergodicity breaking in random media. Phys. Rev. E , 94:052147, 2016

2016
[39]

A. Mura. Non-Markovian Stochastic Processes and Their Applications: From Anomalous Diffusion to Time Series Analysis . Lambert Academic Publishing, 2011. Ph.D. Thesis, Physics Department, University of Bologna, 2008

2011
[40]

Mura and F

A. Mura and F. Mainardi. A class of self-similar stochastic processes with stationary incre- ments to model anomalous diffusion in physics. Integr. Transf. Spec. F. , 20(3–4):185–198, 2009. 34

2009
[41]

Mura and G

A. Mura and G. Pagnini. Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. J. Phys. A: Math. Theor. , 41:285003, 2008

2008
[42]

A. Mura, M. S. Taqqu, and F. Mainardi. Non-Markovian diffusion equations and processes: Analysis and simulations. Physica A, 387:5033–5064, 2008

2008
[43]

Pagnini and P

G. Pagnini and P. Paradisi. A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation. Fract. Calc. Appl. Anal., 19:408–440, 2016

2016
[44]

Runfola and G

C. Runfola and G. Pagnini. Generalized Fokker–Planck equation for superstatistical systems. Physica D, 467:134247, 2024

2024
[45]

Runfola, S

C. Runfola, S. Vitali, and G. Pagnini. The Fokker–Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm. R. Soc. Open Sci., 9:221141, 2022

2022
[46]

Sabri, X

A. Sabri, X. Xu, D. Krapf, and M. Weiss. Elucidating the origin of heterogeneous anomalous diffusion in the cytoplasm of mammalian cells. Phys. Rev. Lett. , 125(5):058101, 2020

2020
[47]

A. A. Sadoon and Y. Wang. Anomalous, non-Gaussian, viscoelastic, and age-dependent dynamics of histonelike nucleoid-structuring proteins in live escherichia coli. Phys. Rev. E , 98:042411, 2018

2018
[48]

R. L. Schilling, R. Song, and Z. Vondraˇ cek. Bernstein functions. Theory and applications , volume 37 of De Gruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin, second edition, 2012

2012
[49]

W. R. Schneider. Grey noise. In Stochastic Processes, Physics and Geometry, pages 676–681. World Scientific, Teaneck, 1990

1990
[50]

W. R. Schneider. Grey noise. In Ideas and Methods in Mathematical Analysis, Stochastics, and Applications, volume I, pages 261–282. Cambridge University Press, Cambridge, 1992

1992
[51]

H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes. Single particle tracking: from theory to biophysical applications. Chem. Rev., 117:7331–7376, 2017

2017
[52]

Simon, L

F. Simon, L. E. Weiss, and S. van Teeffelen. A guide to single-particle tracking. Nat. Rev. Methods Primers, 4:66, 2024

2024
[53]

´Slezak and R

J. ´Slezak and R. Metzler. Minimal model of diffusion with time changing Hurst exponent. J. Phys. A: Math. Theor. , 56:35LT01, 2023

2023
[54]

Speckner and M

K. Speckner and M. Weiss. Single-particle tracking reveals anti-persistent subdiffusion in cell extracts. Entropy, 23(7):892, 2021

2021
[55]

Sposini, A

V. Sposini, A. V. Chechkin, F. Seno, G. Pagnini, and R. Metzler. Random diffusivity from stochastic equations: Comparison of two models for Brownian yet non-Gaussian diffusion. New J. Phys. , 20:043044, 2018. 35

2018
[56]

Sposini, D

V. Sposini, D. S. Grebenkov, R. Metzler, G. Oshanin, and F. Seno. Universal spectral features of different classes of random-diffusivity processes. New J. Phys. , 22:063056, 2020

2020
[57]

Sungkaworn, M

T. Sungkaworn, M. Jobin, K. Burnecki, A. Weron, M. J. Lohse, and D. Calebiro. Single- molecule imaging reveals receptor–g protein interactions at cell surface hot spots. Nature, 550:543–547, 2017

2017
[58]

Szymanski and M

J. Szymanski and M. Weiss. Elucidating the origin of anomalous diffusion in crowded fluids. Phys. Rev. Lett. , 103:038102, 2009

2009
[59]

W. Wang, A. G. Cherstvy, A. V. Chechkin, S. Thapa, F. Seno, X. Liu, and R. Metzler. Fractional Brownian motion with random diffusivity: emerging residual nonergodicity below the correlation time. J. Phys. A: Math. Theor. , 53:474001, 2020

2020
[60]

W. Wang, F. Seno, I. M. Sokolov, A. V. Chechkin, and R. Metzler. Unexpected crossovers in correlated random-diffusivity processes. New J. Phys. , 22:083041, 2020

2020
[61]

M. Weiss. Single-particle tracking data reveal anticorrelated fractional Brownian motion in crowded fluids. Phys. Rev. E , 88:010101(R), 2013

2013
[62]

Woszczek, A

H. Woszczek, A. Wy loma´ nska, and A. Chechkin. Riemann-Liouville fractional Brownian motion with random Hurst exponent. Chaos, 35:023145, 2025. 36

2025

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Balcerek, K

M. Balcerek, K. Burnecki, S. Thapa, A. Wy lomanska, and A. Chechkin. Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions. Chaos, 32(9):093114, 09 2022

work page 2022

[2] [2]

Balcerek, S

M. Balcerek, S. Thapa, K. Burnecki, H. Kantz, R. Metzler, A. Wy loma´ nska, and A. Chechkin. Multifractional Brownian motion with telegraphic, stochastically varying exponent. Phys. Rev. Lett., 134:197101, 2025

work page 2025

[3] [3]

Barkai, Y

E. Barkai, Y. Garini, and R. Metzler. Strange kinetics of single molecules in living cells. Phys. Today, 65(8):29, 2012

work page 2012

[4] [4]

Beghin, L

L. Beghin, L. Cristofaro, and J. Gajda. Non-Gaussian measures in infinite dimensional spaces: the Gamma-grey noise. Potential Anal., 60:1571–1593, 2023

work page 2023

[5] [5]

Bender, M

C. Bender, M. Bormann, and Y. A. Butko. Subordination principle and Feynman–Kac for- mulae for generalized time-fractional evolution equations. Fract. Calc. Appl. Anal., 25:1818– 1836, 2022

work page 2022

[6] [6]

Bender and Y

C. Bender and Y. A. Butko. Stochastic solutions of generalized time-fractional evolution equations. Fract. Calc. Appl. Anal. , 25:488–519, 2022

work page 2022

[7] [7]

Benelli and M

R. Benelli and M. Weiss. From sub-to superdiffusion: Fractional Brownian motion of mem- braneless organelles in early C. elegans embryos. New J. Phys , 23:063072, 2021

work page 2021

[8] [8]

Y. A. Butko and M. Mlinarzik. Fractional Ito calculus for randomly scaled fractional Brow- nian motion and its applications to evolution equations. Preprint, 2024

work page 2024