Fractional Brownian Motion with Negative Hurst Exponent

Baruch Meerson; Pavel V. Sasorov

arxiv: 2507.05977 · v4 · submitted 2025-07-08 · ❄️ cond-mat.stat-mech · math.PR

Fractional Brownian Motion with Negative Hurst Exponent

Baruch Meerson , Pavel V. Sasorov This is my paper

Pith reviewed 2026-05-19 06:04 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math.PR

keywords fractional Brownian motionHurst exponentnegative Hurst exponentstationary Gaussian processanomalous diffusionfractional Ornstein-Uhlenbeck processlong-range correlationsregularization

0 comments

The pith

Regularized fractional Brownian motion with negative Hurst exponents is stationary and suppresses diffusion completely.

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

The paper extends fractional Brownian motion to the regime where the Hurst exponent is negative but greater than -1/2. Using a regularization based on local temporal averaging, the resulting process turns out to be stationary while remaining rough and having long-range positive correlations. Stationarity in this case means that the process does not spread out over time, so diffusion is entirely suppressed. The authors also extend the fractional Ornstein-Uhlenbeck process similarly and find that its smoothed version becomes insensitive to the strength of the confining potential at long times. This matters because it shows how scale-invariant processes behave differently when the Hurst exponent crosses into negative territory, potentially affecting models of persistent fluctuations in physical systems.

Core claim

The central claim is that for Hurst exponents between -1/2 and 0, the regularized fractional Brownian motion is a stationary Gaussian process that is both very rough and persistent due to long-range positive correlations. Because it is stationary, diffusion is completely suppressed. The closely related stationary fractional Ornstein-Uhlenbeck process, when smoothed in the same manner, becomes asymptotically insensitive to the strength of the confining potential. Optimal paths conditioned on reaching a certain value can also be found, and results match continuously with the standard case at the boundary H=0.

What carries the argument

The local temporal averaging with a narrow filter, which regularizes the non-pointwise fractional Brownian motion while preserving its long-range correlation structure and stationarity.

If this is right

Diffusion is completely suppressed due to the stationarity of the regularized process.
The process remains both rough and persistent with long-range positive correlations.
The smoothed fractional Ornstein-Uhlenbeck process becomes asymptotically insensitive to the strength of the confining potential.
Optimal paths for the processes conditioned on reaching a specified value can be determined.
All results connect continuously to the known behaviors of the standard processes at H=0.

Where Pith is reading between the lines

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

The same regularization might extend other Gaussian processes that lack pointwise definitions into new parameter regimes.
Physical systems with long-range temporal correlations could exhibit stationary rather than diffusive behavior in this Hurst range.
Numerical checks of the correlation functions in the smoothed processes could test the claimed persistence and roughness.
The optimal paths may apply to calculating probabilities of rare events in persistent random processes.

Load-bearing premise

A local temporal averaging with a narrow filter provides a mathematically consistent regularization that preserves the long-range correlation structure and stationarity of the underlying process without introducing artifacts.

What would settle it

Simulating the regularized process over long times and checking whether its position variance remains bounded instead of growing linearly or faster would confirm or refute the complete suppression of diffusion.

Figures

Figures reproduced from arXiv: 2507.05977 by Baruch Meerson, Pavel V. Sasorov.

**Figure 2.** Figure 2: FIG. 2. The autocorrelation function [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The variance Var [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Examples of stochastic realizations of the (time-discretized) smoothed processes – the fBm (left panel) and the fGn [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The optimal path [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. The variance of the smoothed fOU process versus [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

read the original abstract

Fractional Brownian motion (fBm) is an important scale-invariant Gaussian non-Markovian process with stationary increments, which serves as a prototypical example of a system with long-range temporal correlations and anomalous diffusion. The fBm is traditionally defined for the Hurst exponent $H$ in the range $0<H<1$. Here we extend this definition to the regime $-1/2<H<0$. The extended fBm is not a pointwise process, so we regularize it via a local temporal averaging with a narrow filter. The resulting process is both very rough and persistent, that is long-range positively correlated. In addition, this process is stationary. The stationarity implies that diffusion is completely suppressed in this region of $H$. We also study another closely related Gaussian process: the stationary fractional Ornstein--Uhlenbeck (fOU) process, extended to the range $-1/2<H<0$ and smoothed in the same way as the fBm. Remarkably, the smoothed fOU process is asymptotically insensitive to the strength of the confining potential. Finally, we determine the optimal paths of the fBm and fOU processes for $-1/2<H<0$, conditioned on reaching a specified value when starting from zero. In the marginal case $H=0$, our results match continuously with known results for the traditionally defined fBm and fOU processes.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

Paper extends fBm and fOU to negative Hurst via regularization, claims stationarity and suppressed diffusion, but regularization needs verification to confirm no distortion of correlations.

read the letter

The main takeaway is that Meerson and Sasorov have pushed the definition of fractional Brownian motion down to negative Hurst exponents between -1/2 and 0. They regularize the non-pointwise process with a narrow local average, and the result turns out to be stationary, rough, and positively correlated over long times. Because it is stationary, the variance stays bounded and ordinary diffusion vanishes. They carry the same regularization over to the fractional Ornstein-Uhlenbeck process. There the smoothed version loses sensitivity to the strength of the restoring force at long times. They also compute the most likely paths that reach a given height starting from zero, and these match onto the known H=0 results. The new ground is the explicit construction for H<0 and the stationarity that follows from it. The paper shows how the covariance can be continued analytically and then smoothed without losing the persistence. That part reads cleanly from the abstract. The potential weak point is whether the chosen filter really leaves the long-range correlations untouched. Any smoothing operation mixes scales, so one has to verify that the large-time covariance keeps its sign and that the process remains exactly stationary once the filter width is sent to zero after other limits. The abstract asserts this, but the details of the filter and the error estimates matter. If those checks are solid, the claims hold; if not, the persistence could be an artifact. This paper sits in the corner of statistical mechanics that deals with Gaussian processes and anomalous transport. Someone already working on long-memory noise or on extensions of fBm would find the negative-H regime useful to know about. It is not a broad overhaul of the field, but it fills a gap that was left open. I think it is worth sending out for refereeing. The core idea is simple and the consequences are stated clearly enough that experts can test the regularization step in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript extends fractional Brownian motion (fBm) from the conventional range 0 < H < 1 to -1/2 < H < 0 via analytic continuation of the covariance kernel, followed by regularization through local temporal averaging with a narrow filter. The resulting process is asserted to be stationary, rough, and long-range positively correlated (persistent), implying complete suppression of diffusion. An analogous extension and smoothing is applied to the fractional Ornstein-Uhlenbeck (fOU) process, which is reported to become asymptotically insensitive to the strength of the confining potential for negative H. Optimal paths conditioned on reaching a target value starting from zero are derived for both processes, with continuous matching to the known positive-H results at the boundary H = 0.

Significance. If the regularization is shown to preserve exact stationarity and the sign of long-range correlations without distorting infrared modes, the work would furnish a concrete realization of stationary rough Gaussian processes with negative Hurst index and suppressed diffusion. This could be relevant for modeling certain persistent yet non-diffusive phenomena in statistical mechanics. The continuous limit at H = 0 and the reported insensitivity of the smoothed fOU are potentially useful features. However, the significance is tempered by the need for explicit verification that the filter does not alter the claimed covariance structure.

major comments (2)

[regularization procedure] The regularization procedure (described after the covariance extension in the main text): the claim that local temporal averaging with finite-width filter yields an exactly stationary process whose covariance depends only on time lag and remains positively correlated at large separations for -1/2 < H < 0 requires an explicit post-filtering calculation of the two-point function. Without showing that the filter width ε can be taken to zero after other limits while preserving the infrared persistence, the stationarity and diffusion-suppression assertions rest on an unverified assumption.
[fractional Ornstein-Uhlenbeck section] fOU extension and potential insensitivity: the statement that the smoothed fOU becomes asymptotically insensitive to the confining potential strength for negative H must be accompanied by a controlled limit (filter width, potential strength, and H) demonstrating that the effective variance or correlation time decouples from the potential parameter; the current outline leaves open whether this holds uniformly or only for specific filter choices.

minor comments (2)

[methods] Clarify the precise definition and width scaling of the 'narrow filter' used for regularization; an explicit functional form or cutoff function would aid reproducibility.
[abstract and conclusions] The abstract states that results 'match continuously' at H=0; a brief remark or plot confirming continuity of key observables (e.g., variance or optimal-path action) at the boundary would strengthen the presentation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for providing constructive feedback. We respond to each of the major comments below, and we will revise the manuscript accordingly to address the points raised.

read point-by-point responses

Referee: The regularization procedure (described after the covariance extension in the main text): the claim that local temporal averaging with finite-width filter yields an exactly stationary process whose covariance depends only on time lag and remains positively correlated at large separations for -1/2 < H < 0 requires an explicit post-filtering calculation of the two-point function. Without showing that the filter width ε can be taken to zero after other limits while preserving the infrared persistence, the stationarity and diffusion-suppression assertions rest on an unverified assumption.

Authors: We agree that an explicit post-filtering calculation is essential to substantiate the claims. In the revised manuscript, we will include a detailed derivation of the two-point function for the smoothed process. This calculation will show that the covariance is a function only of the time lag and that the long-range positive correlations are preserved in the infrared limit when the filter width is taken to zero after the appropriate time limits. This will confirm the stationarity and the resulting suppression of diffusion for -1/2 < H < 0. revision: yes
Referee: fOU extension and potential insensitivity: the statement that the smoothed fOU becomes asymptotically insensitive to the confining potential strength for negative H must be accompanied by a controlled limit (filter width, potential strength, and H) demonstrating that the effective variance or correlation time decouples from the potential parameter; the current outline leaves open whether this holds uniformly or only for specific filter choices.

Authors: We concur that a controlled limit is needed to rigorously establish the insensitivity. In the revision, we will add an analysis of the limiting behavior as the filter width ε approaches zero, for fixed or scaled potential strength, demonstrating that the effective variance and correlation time of the smoothed fOU decouple from the confining potential parameter for negative H. This will show that the property holds in the relevant limit and is robust. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit regularization

full rationale

The paper begins from the standard covariance definition of fBm for H > 0, extends the formal expression to -1/2 < H < 0, and introduces an explicit local temporal averaging regularization to obtain a well-defined process. Stationarity, suppressed diffusion, and persistence are then derived directly from the resulting covariance of the smoothed process. No equations reduce a claimed prediction or property back to a fitted parameter or prior self-citation by construction; the central results follow from the mathematical continuation and smoothing step without circular reduction. The construction is therefore independent of the target claims and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition and covariance of positive-H fBm together with the assumption that local averaging preserves the essential correlation properties while enforcing stationarity.

axioms (2)

standard math Fractional Brownian motion for 0 < H < 1 is a well-defined Gaussian process with stationary increments and the known covariance structure.
Invoked as the starting point for the extension to negative H.
domain assumption Local temporal averaging with a narrow filter produces a well-behaved stationary process whose correlation properties match those implied by the formal negative-H covariance.
Central modeling choice that enables all subsequent claims.

pith-pipeline@v0.9.0 · 5775 in / 1411 out tokens · 59084 ms · 2026-05-19T06:04:30.301386+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We regularize the fBm... via a local temporal averaging with a narrow filter... The resulting process is both very rough and persistent... this process is stationary. The stationarity implies that diffusion is completely suppressed
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

κΔ(τ) = D(2Δ)^{2H} Γ(H+1/2) 1F1(−H;1/2;−τ²/4Δ²)/√π

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

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[1] [1]

A. N. Kolmogorov, C. R. Dokl. Acad. Sci. URSS26, 115 (1940)

work page 1940

[2] [2]

B. B. Mandelbrot and J. W. van Ness, SIAM Review10, 422 (1968)

work page 1968

[3] [3]

Processes with Long-Range Correlations. Theory and Applications

H. Qian, in“Processes with Long-Range Correlations. Theory and Applications”, edited by G. Rangarajan and M. Ding (Springer, Berlin, 2003), p. 22

work page 2003

[4] [4]

W. C. Chow, WIREs Comp. Stat.3, 149 (2011)

work page 2011

[5] [5]

Metzler, J.-H

R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, Phys. Chem. Chem. Phys.16, 24128 (2014)

work page 2014

[6] [6]

Monthus, arXiv:2505.16356

C. Monthus, arXiv:2505.16356

work page arXiv

[7] [7]

Krug, Adv

J. Krug, Adv. Phys.46, 139 (1997)

work page 1997

[8] [8]

N. R. Smith, B. Meerson and P. V. Sasorov, Phys. Rev. E95, 012134 (2017)

work page 2017

[9] [9]

Halpin–Healy and G

T. Halpin–Healy and G. Palasantzas, Europhys. Lett.105, 50001 (2014)

work page 2014

[10] [10]

R. A. L. Almeida, S. O. Ferreira, T. J. Oliveira, and F. D. A. Aar˜ ao Reis, Phys. Rev. B89, 045309 (2014)

work page 2014

[11] [11]

F. D. A. Aar˜ ao Reis, J. Stat. Mech. (2015) P11020

work page 2015

[12] [12]

I. S. S. Carrasco and T. J. Oliveira, Phys. Rev. E93, 012801 (2016)

work page 2016

[13] [13]

Bettelheim and B

E. Bettelheim and B. Meerson, J. Stat. Mech. (2024) 113204

work page 2024

[14] [14]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark,NIST Handbook of Mathematical Functions(Cambridge University Press, Cambridge, UK, 2010)

work page 2010

[15] [15]

Cheredito, H

P. Cheredito, H. Kawaguchi, and M. Maejima, Electron. J. Probab.8, 1 (2003)

work page 2003

[16] [16]

Kaarakka,Fractional Ornstein-Uhlenbeck Processes(Tampere Univ

T. Kaarakka,Fractional Ornstein-Uhlenbeck Processes(Tampere Univ. Technol., Tampere, Finland, 2015). 2 In Ref. [27] a nearly spatially uniform optical driving of colloidal particles was achieved by using 36 laser beams. 11

work page 2015

[17] [17]

Guggenberger, A

T. Guggenberger, A. Chechkin, and R. Metzler, J. Phys. A: Math. Theor.54, 29 (2021)

work page 2021

[18] [18]

Meerson and P

B. Meerson and P. V. Sasorov, J. Phys. A: Math. Theor.57, 445002 (2024)

work page 2024

[19] [19]

C. E. Rasmussen and C. K. I. Williams,Gaussian Processes for Machine Learning(The MIT Press, Boston, 2006)

work page 2006

[20] [20]

Meerson and N

B. Meerson and N. R Smith, J. Phys. A: Math. Theor.52, 415001 (2019)

work page 2019

[21] [21]

Meerson and G

B. Meerson and G. Oshanin, Phys. Rev. E105, 064137 (2022)

work page 2022

[22] [22]

A. K. Hartmann and B. Meerson, Phys. Rev. E109, 014146 (2024)

work page 2024

[23] [23]

Zinn-Justin,Quantum Field Theory and Critical Phenomena, International Series of Monographs on Physics, 4th ed

J. Zinn-Justin,Quantum Field Theory and Critical Phenomena, International Series of Monographs on Physics, 4th ed. (Clarendon, Oxford, UK, 2002)

work page 2002

[24] [24]

Meerson, Phys

B. Meerson, Phys. Rev. E105, 034106 (2022);107, 039902(E) (2023)

work page 2022

[25] [25]

Valov, N

A. Valov, N. Levi, and B. Meerson, Phys. Rev. E110, 024138 (2024)

work page 2024

[26] [26]

F. D. Cunden, P. Facchi, and P. Vivo, J. Phys. A: Math. Theor.49, 135202 (2016)

work page 2016

[27] [27]

G. Geva, T. Admon, M. Levin, and Y. Roichman, Phys. Rev. Lett.134, 218201 (2025)

work page 2025