Confined kinetics and heterogeneous diffusion driven by fractional Gaussian noise: A path integral approach

Cristiane Morais Smith; David Santiago Quevedo; Felipe Segundo Abril-Berm\'udez

arxiv: 2604.10663 · v2 · submitted 2026-04-12 · ❄️ cond-mat.stat-mech · math-ph· math.MP

Confined kinetics and heterogeneous diffusion driven by fractional Gaussian noise: A path integral approach

David Santiago Quevedo , Felipe Segundo Abril-Berm\'udez , Cristiane Morais Smith This is my paper

Pith reviewed 2026-05-13 07:18 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MP

keywords fractional Gaussian noisemultiplicative diffusionpath integralLamperti transformconfined kineticsheterogeneous diffusionFeynman-Kac formula

0 comments

The pith

Multiplicative fractional Gaussian noise and confinement together generate an effective drift that accumulates probability in regions of low noise amplitude.

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

The paper develops a path-integral method for diffusion driven by fractional Gaussian noise that couples multiplicatively to the state. A stationary-phase approximation produces a Gaussian propagator written in terms of the Lamperti transform. In the additive limit this recovers the known path-integral form of fractional Brownian motion and shows equivalence with the Langevin construction. Subordination via the Feynman-Kac formula then yields kinetic equations governed by effective local Hamiltonians. The central demonstration is that multiplicative diffusion combined with confinement induces a drift term whose consequence is probability accumulation where the noise strength is smallest.

Core claim

Using a stationary-phase approximation applied to the path integral, the propagator for a process driven by multiplicative fractional Gaussian noise is obtained as a Gaussian integral in the Lamperti-transformed variable. This recovers the known representation of fractional Brownian motion when the noise is additive and establishes equivalence between the Riemann-Liouville and Langevin constructions. Within the Feynman-Kac framework, subordination to a killing rate leads to a general procedure for deriving kinetic equations expressed through effective local Hamiltonians. The analysis reveals that the combination of multiplicative diffusion and confinement induces an effective drift, causing

What carries the argument

The stationary-phase approximated path integral for the propagator, expressed in the Lamperti-transformed coordinate that converts multiplicative noise into additive noise.

Load-bearing premise

The stationary-phase approximation remains valid for deriving the Gaussian propagator when the multiplicative noise takes a general form.

What would settle it

Numerical simulation of many trajectories from the Langevin equation with multiplicative fractional Gaussian noise inside a confining potential, followed by construction of the steady-state histogram, would show no excess probability density in low-noise regions if the induced-drift claim is incorrect.

read the original abstract

Many complex systems are described by Langevin-type equations in which the noise exhibits long-range correlations and couples to the system in a state-dependent, multiplicative manner, leading to heterogeneous non-Markovian diffusion. Here, we investigate the problem of diffusion driven by fractional Gaussian noise with a general multiplicative coefficient from a path-integral perspective. Using a stationary-phase approximation, we derive a Gaussian propagator expressed in terms of the Lamperti transform of the process. In the additive limit, our results recover the path-integral representation of fractional Brownian motion based on its Riemann-Liouville formulation and establish its equivalence with the Langevin construction. We further analyze the effect of subordinating the process to a killing rate within the Feynman-Kac framework, and develop a general procedure to derive kinetic equations in terms of effective local Hamiltonians. We show that the interplay between multiplicative diffusion and confinement induces an effective drift term, leading to probability accumulation in regions of low noise amplitude.

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

The paper derives an effective drift from multiplicative fractional Gaussian noise under confinement via path integrals and stationary phase, recovering the additive case but leaving the approximation's range unclear.

read the letter

The core result is a path-integral treatment of diffusion driven by fractional Gaussian noise with a general multiplicative prefactor. They apply a stationary-phase approximation to get a Gaussian propagator in terms of the Lamperti transform, then fold in a killing rate through Feynman-Kac to extract effective local Hamiltonians. The payoff is an induced drift that pushes probability toward regions of smaller noise amplitude when the system is confined. That interplay is the new piece they highlight, and it is not just a restatement of prior additive fBm work. They do recover the known Riemann-Liouville path-integral form for the additive limit, which is a useful consistency check. The approach stays within standard frameworks and avoids inventing new operators, so the formal steps look internally coherent on the page. The soft spot is the stationary-phase step itself. For arbitrary multiplicative functions the phase may not stay sharply peaked once the noise correlations are long-range, and the abstract gives no error bound or explicit regime where higher-order terms can be dropped. If that approximation only holds for slowly varying multipliers or weak confinement, the claimed drift and accumulation effect would be narrower than stated. The paper is aimed at people already working on heterogeneous non-Markovian processes in statistical mechanics, especially those who like path-integral or Fokker-Planck routes to effective equations. It is not a broad methodological overhaul, but the specific derivation could be useful for modeling confined systems with state-dependent fractional noise. I would send it to a referee who knows both fractional Brownian motion and path-integral approximations; the derivations are worth checking even if the final claim needs qualification.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a path-integral treatment of diffusion driven by fractional Gaussian noise with a general multiplicative prefactor. A stationary-phase approximation yields a Gaussian propagator expressed via the Lamperti transform of the process. The additive limit recovers the Riemann-Liouville path-integral representation of fractional Brownian motion and establishes equivalence with the Langevin equation. Within the Feynman-Kac framework the authors derive effective local Hamiltonians for confined kinetics and show that the combination of multiplicative diffusion and confinement produces an effective drift, resulting in probability accumulation in regions of low noise amplitude.

Significance. If the central approximation is justified, the work supplies a systematic route to effective kinetic equations for heterogeneous non-Markovian diffusion under confinement, with potential relevance to complex systems. The explicit recovery of the additive fractional-Brownian-motion case and the construction of effective Hamiltonians constitute verifiable strengths.

major comments (1)

[Section deriving the propagator (stationary-phase step)] The stationary-phase approximation used to obtain the Gaussian propagator for arbitrary multiplicative noise (the step that precedes the Lamperti-transform expression and the subsequent Feynman-Kac construction) is presented without an error bound or a stated regime of uniform validity. Because the claimed effective drift and the probability accumulation at low-noise regions rest directly on this propagator, the absence of a quantitative control on the approximation constitutes a load-bearing gap.

minor comments (1)

[Throughout] Notation for the multiplicative coefficient and the killing rate should be introduced once and used consistently; occasional redefinitions obscure the transition from the path-integral to the effective Hamiltonian.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive overall assessment. We address the single major comment below.

read point-by-point responses

Referee: [Section deriving the propagator (stationary-phase step)] The stationary-phase approximation used to obtain the Gaussian propagator for arbitrary multiplicative noise (the step that precedes the Lamperti-transform expression and the subsequent Feynman-Kac construction) is presented without an error bound or a stated regime of uniform validity. Because the claimed effective drift and the probability accumulation at low-noise regions rest directly on this propagator, the absence of a quantitative control on the approximation constitutes a load-bearing gap.

Authors: We agree that an explicit error bound or a clearly stated regime of uniform validity is absent and that this is a substantive gap, given the central role of the propagator in the subsequent derivations. The stationary-phase step is used in the standard semiclassical sense for path integrals of processes with long-range correlations; it is expected to be accurate for moderate noise strengths and long times, but we did not quantify the remainder. In the revised manuscript we will add a dedicated paragraph (new subsection 2.3) that (i) recalls the conditions under which the stationary-phase approximation is controlled for Gaussian actions, (ii) states the leading-order character of the result, and (iii) supplies direct numerical comparisons between the analytic propagator and Monte-Carlo trajectories of the underlying multiplicative fractional-Gaussian-noise SDE for representative multiplicative prefactors. These additions will make the domain of applicability explicit and will allow readers to assess the accuracy of the effective drift and accumulation predictions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The derivation proceeds from the path-integral representation of multiplicative fGn, applies a stationary-phase approximation to obtain a Gaussian propagator expressed via the Lamperti transform, recovers the known additive fBm case through the external Riemann-Liouville formulation, and then invokes the standard Feynman-Kac formula to introduce a killing rate and extract effective local Hamiltonians. None of these steps reduce by construction to their own inputs, fitted parameters renamed as predictions, or load-bearing self-citations; the effective drift and probability accumulation at low-noise regions emerge as consequences of the derived equations rather than being presupposed. The paper remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical tools from stochastic processes and path integrals without introducing new free parameters or entities.

axioms (2)

domain assumption The stationary-phase approximation can be applied to the path integral formulation of the diffusion process.
This is used to obtain the Gaussian propagator.
domain assumption The Lamperti transform linearizes the multiplicative noise process.
Central to expressing the propagator.

pith-pipeline@v0.9.0 · 5479 in / 1187 out tokens · 114648 ms · 2026-05-13T07:18:17.466361+00:00 · methodology

Review history (2 revisions) →

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

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

Using a stationary-phase approximation, we derive a Gaussian propagator expressed in terms of the Lamperti transform of the process... ∂t P − H t^{2H−1} ∂x [a(x) ∂x [a(x) P]] = 0
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

the interplay between multiplicative diffusion and confinement induces an effective drift term, leading to probability accumulation in regions of low noise amplitude

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

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[1] [1]

Mandelbrot, Self-affinity and fractal dimension, Phys

B. Mandelbrot, Self-affinity and fractal dimension, Phys. Scr.32, 257 (1985)

work page 1985

[2] [2]

Gneiting and M

T. Gneiting and M. Schlather, Stochastic models that separate fractal dimension and the Hurst effect, SIAM Rev.46, 269 (2004)

work page 2004

[3] [3]

H. E. Hurst, Long-term storage capacity of reservoirs, Trans. Am. Soc. Civil Eng.116, 770 (1951)

work page 1951

[4] [4]

A. N. Kolmogorov, Wienersche spiralen und einige andere 6 interessante kurven in hilbertscen raum, cr (doklady), Acad. Sci. URSS (NS)26, 115 (1940)

work page 1940

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P. L´ evy,Random functions: general theory with special reference to Laplacian random functions, Vol. 1 (Univer- sity of California Press. California, USA, 1953)

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W. Wang, Q. Wei, A. V. Chechkin, and R. Metzler, Dif- ferent behaviors of diffusing diffusivity dynamics based on three different definitions of fractional brownian mo- tion, Phys. Rev. E112, 014108 (2025)

work page 2025

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Calvo and R

I. Calvo and R. Sanchez, The path integral formulation of fractional Brownian motion for the general Hurst ex- ponent, J. Phys. A: Math. Theor.41, 282002 (2008)

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Sebastian, Path integral representation for fractional brownian motion, Journal of Physics A: Mathematical and General28, 4305 (1995)

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B. Meerson, O. B´ enichou, and G. Oshanin, Path integrals for fractional Brownian motion and fractional Gaussian noise, Phys. Rev. E106, L062102 (2022)

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Joo and J.-H

S. Joo and J.-H. Jeon, Viscoelastic active diffusion gov- erned by nonequilibrium fractional Langevin equations: Underdamped dynamics and ergodicity breaking, Chaos, Solitons Fractals177, 114288 (2023)

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X. Durang, C. Lim, and J.-H. Jeon, Generalized langevin equation for a tagged monomer in a gaussian semiflexible polymer, J. Chem. Phys.161(2024)

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A. G. Cherstvy, W. Wang, R. Metzler, and I. M. Sokolov, Inertia triggers nonergodicity of fractional brownian mo- tion, Phys. Rev. E104, 024115 (2021)

work page 2021

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Quevedo, R

D. Quevedo, R. Verstraten, and C. Morais Smith, Emer- gent transient time crystal from a fractional Langevin equation with white and colored noise, Phys. Rev. A110, 052208 (2024)

work page 2024

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J. R. Gomez-Solano and F. J. Sevilla, Active particles with fractional rotational Brownian motion, Jour. Stat. Mech: Theory Exp.2020, 063213 (2020)

work page 2020

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D. S. Quevedo, M. Conte, M. Dijkstra, and C. M. Smith, Active brownian particles in power-law viscoelastic me- dia, arXiv preprint arXiv:2512.20205 (2025)

work page arXiv 2025

[24] [24]

S. M. J. Khadem, R. Klages, and S. H. Klapp, Stochas- tic thermodynamics of fractional brownian motion, Phys. Rev. Res.4, 043186 (2022)

work page 2022

[25] [25]

Squarcini, A

A. Squarcini, A. Solon, P. Viot, and G. Oshanin, Frac- tional brownian gyrator, J. Phys. A: Math. Theor.55, 485001 (2022)

work page 2022

[26] [26]

Giordano, F

S. Giordano, F. Cleri, and R. Blossey, Infinite ergodicity in generalized geometric brownian motions with nonlin- ear drift, Phys. Rev. E107, 044111 (2023)

work page 2023

[27] [27]

Abril-Berm´ udez, C

F. Abril-Berm´ udez, C. Quimbay, J. Trinidad-Segovia, and M. S´ anchez-Granero, Path integral for multiplicative noise: Generalized Fokker-Planck equation and entropy production rate in stochastic processes with threshold, Phys. Rev. Res.7, 023185 (2025)

work page 2025

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A. G. Cherstvy, A. V. Chechkin, and R. Metzler, Anoma- lous diffusion and ergodicity breaking in heterogeneous diffusion processes, New J. Phys.15, 083039 (2013)

work page 2013

[30] [30]

W. Wang, A. G. Cherstvy, X. Liu, and R. Metzler, Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise, Phys. Rev. E102, 012146 (2020)

work page 2020

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Lamperti, Semi-stable stochastic processes, Trans

J. Lamperti, Semi-stable stochastic processes, Trans. Am. Math. Soc.104, 62 (1962)

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P. H¨ anggi, Path integral solutions for non-markovian pro- cesses, Z. Phys. B: Condens. Matter75, 275 (1989)

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Parisi and N

G. Parisi and N. Sourlas, Random magnetic fields, su- persymmetry, and negative dimensions, Phys. Rev. Lett. 43, 744 (1979)

work page 1979

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Balaji, Universal nonlinear filtering using feynman path integrals ii: the continuous-continuous model with additive noise, PMC Phys

B. Balaji, Universal nonlinear filtering using feynman path integrals ii: the continuous-continuous model with additive noise, PMC Phys. A3, 2 (2009)

work page 2009

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Onsager and S

L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev.91, 1505 (1953)

work page 1953

[38] [38]

Machlup and L

S. Machlup and L. Onsager, Fluctuations and irreversible process. ii. systems with kinetic energy, Phys. Rev.91, 1512 (1953)

work page 1953

[39] [39]

To our knowledge, such generalized version of the CTRW has not been studied yet in the literature

work page

[40] [40]

Hilfer and L

R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Physical Review E51, R848 (1995)

work page 1995

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