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arxiv: 2604.15952 · v1 · submitted 2026-04-17 · ❄️ cond-mat.stat-mech

Ergodic properties of functionals of Gaussian processes

Vicen\c{c} M\'endez , Carlos Herv\'as , Rosa Flaquer-Galm\'es This is my paper

Pith reviewed 2026-05-10 07:13 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords ergodicityoccupation timeGaussian random walkfractional Brownian motionstochastic functionalsinfinite ergodic theoryergodicity breaking
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0 comments X

The pith

Positive functionals of stationary Gaussian processes have their first two moments determined exactly by the one- and two-time probability densities.

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

The paper derives a general way to obtain the mean and variance of any positive observable made from a stationary random walk by integrating against its single-time and two-time probability densities. This matters for quantities such as the fraction of time a walker spends in a region, because those observables appear throughout physics yet their statistics are often hard to compute directly. The derivation includes a proof that the observables are ergodic whenever the underlying walk is stationary, so that long-time averages coincide with ensemble averages. The same framework yields closed-form moment expressions for half-occupation and interval occupation times of ordinary Gaussian walks and is then extended to scaled Brownian motion and fractional Brownian motion.

Core claim

We derive the first two moments of generic positive stochastic functionals in terms of the one- and two-time probability density functions of the underlying random walk, and we prove ergodicity of observables in stationary random walks. These general results are applied to the half-occupation time and the occupation time in an interval of a Gaussian random walk, for which we obtain exact analytic expressions for the first two moments. We then extend the analysis to scaled Brownian motion and fractional Brownian motion, computing the ergodicity breaking parameter and establishing a simple scaling form for the probability densities of occupation times. Within the framework of infinite ergodic

What carries the argument

Reduction of the moments of a positive functional to integrals over the one- and two-time probability densities of the underlying stationary process.

If this is right

  • Exact analytic expressions exist for the first two moments of half-occupation time and interval occupation time in Gaussian random walks.
  • An ergodicity breaking parameter can be computed for scaled Brownian motion and fractional Brownian motion.
  • Probability densities of occupation times obey a simple scaling form in these processes.
  • Positive observables share universal properties under infinite ergodic theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction may apply to other positive path functionals such as the time integral of a nonlinear function of the position.
  • Relaxing stationarity would require time-dependent densities and would likely produce non-ergodic behavior whose parameter dependence remains to be mapped.
  • The scaling forms found for fractional Brownian motion suggest analogous reductions could be tested in other anomalous-diffusion models that admit two-time densities.

Load-bearing premise

The moments of the positive functionals are fully determined by the one- and two-time probability densities when the underlying process is a stationary Gaussian random walk or a related fractional Brownian motion.

What would settle it

A direct Monte Carlo sampling of the first two moments of the half-occupation time for a stationary Gaussian walk that deviates from the closed-form expressions obtained by integrating the one- and two-time densities.

Figures

Figures reproduced from arXiv: 2604.15952 by Carlos Herv\'as, Rosa Flaquer-Galm\'es, Vicen\c{c} M\'endez.

Figure 1
Figure 1. Figure 1: FIG. 1. First -a)- and second -b)- moments of the half occupation time over [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. First -a)- and second -b)- moments of the occupation time in an interval as a function [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Ergodicity breaking parameter EB as function of the exponent [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. First -a)- and second -b)- moments of the half occupation time over [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. First -a)- and second -b)- moments of the occupation time in an interval as a function of [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Ergodicity breaking parameter EB as function of the exponent [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Scaling form for [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Scaling form for [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Quotient [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
read the original abstract

We derive the first two moments of generic positive stochastic functionals in terms of the one- and two-time probability density functions of the underlying random walk, and we prove ergodicity of observables in stationary random walks. These general results are applied to the half-occupation time and the occupation time in an interval of a Gaussian random walk, for which we obtain exact analytic expressions for the first two moments. We then extend the analysis to scaled Brownian motion and fractional Brownian motion, computing the ergodicity breaking parameter and establishing a simple scaling form for the probability densities of occupation times. Within the framework of infinite ergodic theory, we further identify universal properties of positive observables. All analytical predictions are fully confirmed by numerical simulations.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The manuscript derives the first two moments of generic positive stochastic functionals of random walks in terms of the one- and two-time probability density functions of the underlying process. It proves ergodicity of observables for stationary random walks and applies the framework to the half-occupation time and occupation time in an interval for Gaussian random walks, obtaining exact analytic expressions for the moments. The analysis is extended to scaled Brownian motion and fractional Brownian motion, yielding the ergodicity breaking parameter, a scaling form for the occupation-time PDFs, and universal properties identified via infinite ergodic theory. All analytic predictions are stated to be confirmed by numerical simulations.

Significance. If the derivations are correct, the work supplies explicit analytic results for moments of occupation functionals in Gaussian and self-similar processes, which are relevant for quantifying ergodicity breaking in anomalous diffusion. The general moment expressions and the infinite-ergodic-theory discussion provide a systematic route to positive observables, while the numerical agreement supports the closed-form claims for the specific cases treated.

major comments (2)
  1. [general derivation (prior to specific applications)] The general derivation of the first two moments from one- and two-time PDFs follows directly from the integral definitions of the expectations; the manuscript should clarify the novel technical step beyond this identity, especially since the ergodicity statement for stationary walks follows from the standard Birkhoff ergodic theorem.
  2. [section on fBM and infinite ergodic theory] For the extension to fractional Brownian motion and infinite ergodic theory, the universal properties of positive observables require an explicit statement of the invariant measure and the class of observables for which the scaling form of the PDFs holds; without this, the claim that the properties are universal remains incompletely supported.
minor comments (3)
  1. The numerical simulations confirming the analytic expressions should report the number of trajectories, discretization scheme, and statistical error estimates to permit independent verification.
  2. Notation for the one- and two-time PDFs should be introduced consistently and distinguished from the occupation-time densities to avoid ambiguity in the integral expressions.
  3. A brief comparison with existing results on occupation times for Brownian motion (e.g., Lévy arcsine law and its generalizations) would help situate the new analytic expressions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the positive assessment of the work's significance and the recommendation for minor revision. Below we provide point-by-point responses to the major comments, indicating the revisions we will implement.

read point-by-point responses

  1. Referee: [general derivation (prior to specific applications)] The general derivation of the first two moments from one- and two-time PDFs follows directly from the integral definitions of the expectations; the manuscript should clarify the novel technical step beyond this identity, especially since the ergodicity statement for stationary walks follows from the standard Birkhoff ergodic theorem.

    Authors: We thank the referee for highlighting this point. The moment expressions do follow from direct integration, but the technical contribution of the general framework is to express the first two moments of arbitrary positive functionals solely in terms of the one- and two-time PDFs of the underlying process. This formulation is what subsequently permits closed-form analytic results for the specific occupation-time functionals without invoking higher-order statistics. For the ergodicity statement, we apply the Birkhoff theorem to stationary walks but explicitly verify its consequences for the positive observables under consideration. We will revise the general derivation section to clarify these aspects and distinguish the identity from the subsequent applications. revision: partial

  2. Referee: [section on fBM and infinite ergodic theory] For the extension to fractional Brownian motion and infinite ergodic theory, the universal properties of positive observables require an explicit statement of the invariant measure and the class of observables for which the scaling form of the PDFs holds; without this, the claim that the properties are universal remains incompletely supported.

    Authors: We agree that an explicit statement strengthens the universality claim. Within the infinite ergodic theory framework applied here, the relevant object is the infinite invariant measure associated with the null-recurrent dynamics of the underlying Gaussian process, and the scaling form of the occupation-time PDFs holds for the class of positive observables that possess a finite mean with respect to this measure. We will add a concise paragraph in the fractional Brownian motion section stating the invariant measure (with reference to the relevant infinite ergodic theory results) and the precise conditions on the observables, thereby supporting the universality statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is definitional and self-contained

full rationale

The central results express the first two moments of positive functionals directly as integrals over the one- and two-time PDFs of the underlying process. This follows immediately from the definition of expectation for any stochastic process and does not constitute a non-trivial derivation that reduces to its own inputs. For the Gaussian and fBM cases the bivariate densities are known in closed form, permitting explicit integration without parameter fitting or self-referential steps. Ergodicity claims for stationary walks invoke standard Birkhoff arguments, while extensions to infinite ergodic theory follow from scaling already present in the moment expressions. No self-citation is load-bearing, no ansatz is smuggled, and no uniqueness theorem is invoked from prior author work. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions about Gaussian and fractional Brownian processes and on results from infinite ergodic theory; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption One- and two-time probability density functions of stationary Gaussian random walks fully determine moments of positive functionals
    Invoked to obtain exact analytic expressions for occupation times.
  • domain assumption Results from infinite ergodic theory apply to positive observables of these processes
    Used to identify universal properties.

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Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages

  1. [1]

    Half occupation time We consider here the half occupation timeT +(t) for the SBM. From (52) and (63) T +(t)2 = t2 2 − 1 π Z t 0 dτ2 Z τ2 0 dτ1 arctan s τ α 2 τ α 1 −1 ! = t2 2 " 1− 1 π Z 1 0 arctan r 1 uα −1 ! du # .(64) The integral in the above expression can be solved by introducing the new variabley= (u−α −1) 1/2 Z 1 0 arctan r 1 uα −1 ! du= 2 α Z ∞ 0...

    work page

  2. [2]

    Occupation time in an interval 102 103 104 105 106 t 101 102 103 104 Ta(t) a) =0.5 =1.0 =1.5 102 103 104 105 106 t 102 104 106 108 Ta(t)2 b) =0.5 =1.0 =1.5 FIG. 2. First -a)- and second -b)- moments of the occupation time in an interval as a function oftfor the SBM. The symbols are obtained from numerical simulations forα= 0.5,1.0,and 1.5. The solid lines...

    work page

  3. [3]

    Half occupation time 102 103 104 105 106 t 0.40 0.45 0.50 0.55 0.60 T + (t) /t a) H=0.2 H=0.4 H=0.6 H=0.8 102 103 104 105 106 t 0.30 0.35 0.40 0.45 0.50 T + (t)2 /t2 b) H=0.2 H=0.4 H=0.6 H=0.8 FIG. 4. First -a)- and second -b)- moments of the half occupation time overtandt 2 respectively as a function oftfor the fBM. The symbols are obtained from numerica...

    work page

  4. [4]

    Occupation time in an interval 102 103 104 105 106 t 101 102 103 104 105 Ta(t) a) H=0.2 H=0.4 H=0.6 H=0.8 102 103 104 105 106 t 104 106 108 1010 Ta(t)2 b) H=0.2 H=0.4 H=0.6 H=0.8 FIG. 5. First -a)- and second -b)- moments of the occupation time in an interval as a function of tfor the fBM. The symbols are obtained from numerical simulations forH= 0.2,0.4,...

    work page

  5. [5]

    erf a− ⟨x(τ)⟩p 2C(τ, τ) ! + erf a+⟨x(τ)⟩p 2C(τ, τ) !# (D2) where the integration constant is zero provided thatI 2(a= 0) = 0. Finally, Eq. (48) has the form ⟨Ta(t)⟩= 1 2 Z t 0

    (which we include in the plot as orange squares with the label KA), the EB a computed from the ML distribution is valid solely in the region nearH= 1/2, where Eq. (76) and (70) provide similar results. For values ofH→0 andH→1 we find that (70) is no longer valid. However, our result (76) is exact and holds for the whole range of values ofH. 0.2 0.4 0.6 0....

    work page

  6. [6]

    Kac, On distributions of certain wiener functionals, Transactions of the American Mathe- matical Society65, 1 (1949)

    M. Kac, On distributions of certain wiener functionals, Transactions of the American Mathe- matical Society65, 1 (1949)

    work page 1949

  7. [7]

    S. N. Majumdar, Brownian Functionals in Physics and Computer Science, Current Science 89, 2076 (2005)

    work page 2076

  8. [8]

    Kac, On Some Connections between Probability Theory and Differential and Integral Equations, inSecond Berkeley Symposium on Mathematical Statistics and Probability, edited by J

    M. Kac, On Some Connections between Probability Theory and Differential and Integral Equations, inSecond Berkeley Symposium on Mathematical Statistics and Probability, edited by J. Neyman (1951) pp. 189–215

    work page 1951

  9. [9]

    Yor, Some aspects of brownian motion

    M. Yor, Some aspects of brownian motion. lectures in math., eth z urich (1992)

    work page 1992

  10. [10]

    Geman and M

    H. Geman and M. Yor, Bessel processes, asian options, and perpetuities, Mathematical Finance 3, 349 (1993). 33

    work page 1993

  11. [11]

    Comtet, J

    A. Comtet, J. Desbois, and C. Texier, Functionals of brownian motion, localization and metric graphs, Journal of Physics A: Mathematical and General38, R341 (2005)

    work page 2005

  12. [12]

    Baule and R

    A. Baule and R. Friedrich, Investigation of a generalized obukhov model for turbulence, Physics Letters A350, 167 (2006)

    work page 2006

  13. [13]

    S. N. Majumdar and A. J. Bray, Large-deviation functions for nonlinear functionals of a gaussian stationary markov process, Physical Review E65, 051112 (2002)

    work page 2002

  14. [14]

    Agmon, Residence times in diffusion processes, The Journal of Chemical Physics81, 3644 (1984)

    N. Agmon, Residence times in diffusion processes, The Journal of Chemical Physics81, 3644 (1984)

    work page 1984

  15. [15]

    Agmon, The residence time equation, Chemical Physics Letters497, 184 (2010)

    N. Agmon, The residence time equation, Chemical Physics Letters497, 184 (2010)

    work page 2010

  16. [16]

    Godreche and J.-M

    C. Godreche and J.-M. Luck, Statistics of the occupation time of renewal processes, Journal of Statistical Physics104, 489–524 (2001)

    work page 2001

  17. [17]

    L´ evy, Sur certains processus stochastiques homog` enes, Compositio Mathematica7, 283 (1940)

    P. L´ evy, Sur certains processus stochastiques homog` enes, Compositio Mathematica7, 283 (1940)

    work page 1940

  18. [18]

    Lamperti, An occupation time theorem for a class of stochastic processes, Transactions of the American Mathematical Society88, 380 (1958)

    J. Lamperti, An occupation time theorem for a class of stochastic processes, Transactions of the American Mathematical Society88, 380 (1958)

    work page 1958

  19. [19]

    Singh and A

    P. Singh and A. Kundu, Generalised ‘arcsine’ laws for run-and-tumble particle in one dimen- sion, Journal of Statistical Mechanics: Theory and Experiment2019, 083205 (2019)

    work page 2019

  20. [20]

    Singh, Extreme value statistics and arcsine laws for heterogeneous diffusion processes, Physical Review E105, 024113 (2022)

    P. Singh, Extreme value statistics and arcsine laws for heterogeneous diffusion processes, Physical Review E105, 024113 (2022)

    work page 2022

  21. [21]

    S. N. Majumdar and A. Comtet, Local and occupation time of a particle diffusing in a random medium, Physical Review Letters89, 060601 (2002)

    work page 2002

  22. [22]

    Sabhapandit, S

    S. Sabhapandit, S. N. Majumdar, and A. Comtet, Statistical properties of functionals of the paths of a particle diffusing in a one-dimensional random potential, Physical Review E73, 051102 (2006)

    work page 2006

  23. [23]

    Barkai, Residence time statistics for normal and fractional diffusion in a force field, Journal of Statistical Physics123, 883 (2006)

    E. Barkai, Residence time statistics for normal and fractional diffusion in a force field, Journal of Statistical Physics123, 883 (2006)

    work page 2006

  24. [24]

    Turgeman, S

    L. Turgeman, S. Carmi, and E. Barkai, Fractional feynman-kac equation for non-brownian functionals, Physical Review Letters103, 190201 (2009)

    work page 2009

  25. [25]

    Carmi, L

    S. Carmi, L. Turgeman, and E. Barkai, On distributions of functionals of anomalous diffusion paths, Journal of Statistical Physics141, 1071 (2010). 34

    work page 2010

  26. [27]

    Y. He, S. Burov, R. Metzler, and E. Barkai, Random time-scale invariant diffusion and trans- port coefficients, Physical Review Letters101, 058101 (2008)

    work page 2008

  27. [28]

    Thiel and I

    F. Thiel and I. M. Sokolov, Weak ergodicity breaking in an anomalous diffusion process of mixed origins, Physical Review E89, 012136 (2014)

    work page 2014

  28. [29]

    Meroz and I

    Y. Meroz and I. M. Sokolov, A toolbox for determining subdiffusive mechanisms, Physics Reports573, 1 (2015)

    work page 2015

  29. [30]

    Burov, R

    S. Burov, R. Metzler, and E. Barkai, Aging and nonergodicity beyond the khinchin theorem, Proceedings of the National Academy of Sciences107, 13228 (2010)

    work page 2010

  30. [31]

    Barkai, R

    E. Barkai, R. Flaquer-Galm´ es, and V. M´ endez, Ergodic properties of brownian motion under stochastic resetting, Physical Review E108, 064102 (2023)

    work page 2023

  31. [32]

    Dhar and S

    A. Dhar and S. N. Majumdar, Residence time distribution for a class of gaussian markov processes, Physical Review E59, 6413 (1999)

    work page 1999

  32. [33]

    Von Neumann,Invariant measures(American Mathematical Soc

    J. Von Neumann,Invariant measures(American Mathematical Soc. Providence, Rhode Island, 1941)

    work page 1941

  33. [34]

    Leibovich and E

    N. Leibovich and E. Barkai, Infinite ergodic theory for heterogeneous diffusion processes, Physical Review E99, 042138 (2019)

    work page 2019

  34. [35]

    M´ endez, R

    V. M´ endez, R. Flaquer-Galm´ es, and A. Pal, Occupation-time statistics for non-markovian random walks, Physical Review E111, 044119 (2025)

    work page 2025

  35. [36]

    Khinchin,Mathematical Foundations of Statistical Mechanics(Dover, New York, 1949)

    A. Khinchin,Mathematical Foundations of Statistical Mechanics(Dover, New York, 1949)

    work page 1949

  36. [37]

    N. G. Van Kampen,Stochastic processes in physics and chemistry, Vol. 1 (Elsevier, 1992)

    work page 1992

  37. [38]

    Metzler, J.-H

    R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Physical Chemistry Chemical Physics16, 24128 (2014)

    work page 2014

  38. [39]

    M. J. Saxton, Anomalous subdiffusion in fluorescence photobleaching recovery: A monte carlo study, Biophysical Journal81, 2226 (2001)

    work page 2001

  39. [40]

    Szyma´ nski, A

    J. Szyma´ nski, A. Patkowski, J. Gapi´ nski, A. Wilk, and R. Ho lyst, Movement of proteins in an environment crowded by surfactant micelles: Anomalous versus normal diffusion, The Journal of Physical Chemistry B110, 7367 (2006). 35

    work page 2006

  40. [41]

    Wu and K

    J. Wu and K. M. Berland, Propagators and time-dependent diffusion coefficients for anomalous diffusion, Biophysical Journal95, 2049 (2008)

    work page 2049

  41. [42]

    Kazakeviˇ cius and A

    R. Kazakeviˇ cius and A. Kononovicius, Anomalous diffusion and long-range memory in the scaled voter model, Physical Review E107, 024106 (2023)

    work page 2023

  42. [43]

    L. F. Richardson, Atmospheric diffusion shown on a distance-neighbour graph, Proceedings of the Royal Society A110, 709 (1926)

    work page 1926

  43. [44]

    A. S. Bodrova, A. V. Chechkin, A. G. Cherstvy, and R. Metzler, Ultraslow scaled brownian motion, New Journal of Physics17, 063038 (2015)

    work page 2015

  44. [45]

    Safdari, A

    H. Safdari, A. G. Cherstvy, A. V. Chechkin, F. Thiel, I. M. Sokolov, and R. Metzler, Quanti- fying the non-ergodicity of scaled brownian motion, Journal of Physics A: Mathematical and Theoretical48, 375002 (2015)

    work page 2015

  45. [46]

    J.-H. Jeon, A. V. Chechkin, and R. Metzler, Scaled brownian motion: a paradoxical pro- cess with a time dependent diffusivity for the description of anomalous diffusion, Physical Chemistry Chemical Physics16, 15811 (2014)

    work page 2014

  46. [47]

    A. G. Cherstvy and R. Metzler, Ergodicity breaking, ageing, and confinement in general- ized diffusion processes with position and time dependent diffusivity, Journal of Statistical Mechanics: Theory and Experiment2015, P05010 (2015)

    work page 2015

  47. [48]

    D. A. Darling and M. Kac, On occupation times for markoff processes, Transactions of the American Mathematical Society84, 444 (1957)

    work page 1957

  48. [49]

    Kimura and T

    M. Kimura and T. Akimoto, Occupation time statistics of the fractional brownian motion in a finite domain, Physical Review E106, 064132 (2022)

    work page 2022

  49. [50]

    Safdari, A

    H. Safdari, A. V. Chechkin, G. R. Jafari, and R. Metzler, Aging scaled brownian motion, Physical Review E91, 042107 (2015)

    work page 2015

  50. [51]

    A. N. Kolmogorov, Wiener’s spiral and some other interesting curves in hilbert space, Comptes Rendus de l’Acad´ emie des sciences de l’URSS26, 115 (1940)

    work page 1940

  51. [52]

    B. B. Mandelbrot and J. W. Van Ness, Fractional brownian motions, fractional noises and applications, SIAM review10, 422 (1968)

    work page 1968

  52. [53]

    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, Physical Review Letters103, 180602 (2009). 36

    work page 2009

  53. [54]

    J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Odd- ershede, and R. Metzler, In vivo anomalous diffusion and weak ergodicity breaking of lipid granules, Physical Review Letters106, 048103 (2011)

    work page 2011

  54. [55]

    J.-H. Jeon, H. M.-S. Monne, M. Javanainen, and R. Metzler, Anomalous diffusion of phospho- lipids and cholesterols in a lipid bilayer and its origins, Physical Review Letters109, 188103 (2012)

    work page 2012

  55. [56]

    J.-H. Jeon, N. Leijnse, L. B. Oddershede, and R. Metzler, Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions, New Journal of Physics15, 045011 (2013)

    work page 2013

  56. [57]

    S. A. Tabei, S. Burov, H. Y. Kim, A. Kuznetsov, T. Huynh, J. Jureller, L. H. Philipson, A. R. Dinner, and N. F. Scherer, Intracellular transport of insulin granules is a subordinated random walk, Proceedings of the National Academy of Sciences110, 4911 (2013)

    work page 2013

  57. [58]

    Mikosch, S

    T. Mikosch, S. Resnick, H. Rootz´ en, and A. Stegeman, Is network traffic appriximated by stable l´ evy motion or fractional brownian motion?, The annals of applied probability12, 23 (2002)

    work page 2002

  58. [59]

    Comte and E

    F. Comte and E. Renault, Long memory in continuous-time stochastic volatility models, Math- ematical Finance8, 291 (1998)

    work page 1998

  59. [60]

    Biagini, Y

    F. Biagini, Y. Hu, B. Øksendal, and T. Zhang,Stochastic calculus for fractional Brownian motion and applications(Springer, 2008)

    work page 2008

  60. [61]

    J. F. Reverey, J.-H. Jeon, H. Bao, M. Leippe, R. Metzler, and C. Selhuber-Unkel, Superdif- fusion dominates intracellular particle motion in the supercrowded cytoplasm of pathogenic acanthamoeba castellanii, Scientific reports5, 11690 (2015)

    work page 2015

  61. [62]

    Krapf, N

    D. Krapf, N. Lukat, E. Marinari, R. Metzler, G. Oshanin, C. Selhuber-Unkel, A. Squarcini, L. Stadler, M. Weiss, and X. Xu, Spectral content of a single non-brownian trajectory, Physical Review X9, 011019 (2019)

    work page 2019

  62. [63]

    Rangarajan and M

    G. Rangarajan and M. Ding,Processes with long-range correlations: Theory and applications, Vol. 621 (Springer, 2008)

    work page 2008

  63. [64]

    Deng and E

    W. Deng and E. Barkai, Ergodic properties of fractional brownian-langevin motion, Physical Review E79, 011112 (2009)

    work page 2009

  64. [65]

    Jeon and R

    J.-H. Jeon and R. Metzler, Fractional brownian motion and motion governed by the fractional langevin equation in confined geometries, Physical Review E81, 021103 (2010). 37

    work page 2010

  65. [66]

    Sadhu, M

    T. Sadhu, M. Delorme, and K. J. Wiese, Generalized arcsine laws for fractional brownian motion, Physical Review Letters120, 040603 (2018)

    work page 2018

  66. [67]

    K. J. Wiese, S. N. Majumdar, and A. Rosso, Perturbation theory for fractional brownian motion in presence of absorbing boundaries, Physical Review E83, 061141 (2011)

    work page 2011

  67. [68]

    Vojta and A

    T. Vojta and A. Warhover, Probability density of fractional brownian motion and the frac- tional langevin equation with absorbing walls, Journal of Statistical Mechanics: Theory and Experiment2021, 033215 (2021)

    work page 2021

  68. [69]

    Perrin, R

    E. Perrin, R. Harba, R. Jennane, and I. Iribarren, Fast and exact synthesis for 1-d fractional brownian motion and fractional gaussian noises, IEEE Signal Processing Letters9, 382 (2002)

    work page 2002

  69. [70]

    W. Wang, R. Metzler, and A. G. Cherstvy, Anomalous diffusion, aging, and nonergodicity of scaled brownian motion with fractional gaussian noise: overview of related experimental observations and models, Physical Chemistry Chemical Physics24, 18482 (2022). 38

    work page 2022