Second- and third-order properties of multidimensional Langevin equations

McGill University); Yeeren I. Low (Department of Physics

arxiv: 2312.04585 · v17 · submitted 2023-12-04 · ❄️ cond-mat.stat-mech · physics.data-an· q-bio.QM

Second- and third-order properties of multidimensional Langevin equations

Yeeren I. Low (Department of Physics , McGill University) This is my paper

Pith reviewed 2026-05-24 05:11 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.data-anq-bio.QM

keywords Langevin equationstochastic processesprobability density momentsprobability currentMarkovian limitnon-Markovian detectionunderdamped dynamics

0 comments

The pith

Terms in multidimensional Langevin equations correspond to moments of the probability density and current density even for nonlinear and underdamped dynamics.

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

This paper establishes connections between the coefficients of a Langevin equation and various statistical properties of the system, including moments of the probability density function, the probability current density, and covariance functions. It reviews these relations for linear Gaussian dynamics before extending them to cases with nonlinear drift or diffusion and to underdamped processes that are nearly Markovian. The analysis includes methods to quantify the importance of higher-order effects and to identify signatures of non-Markovian behavior. These relations matter because they allow one to extract interpretable physical parameters from observed statistics without restricting to simple Gaussian processes.

Core claim

The drift vector and diffusion matrix in a multidimensional Langevin equation generate explicit expressions for the time evolution of moments of the probability density, the divergence of the probability current, and the covariance matrix; these expressions continue to hold with measurable quantitative contributions when the dynamics leave the linear Gaussian regime or when the process is underdamped but close to the Markovian limit.

What carries the argument

The explicit mapping from Langevin drift and diffusion terms to the time derivatives of probability moments and currents

If this is right

Nonlinear terms in the drift produce measurable contributions to third-order moments of the probability density.
Diffusion coefficients can be recovered from second moments even in the presence of nonlinearities.
In the almost-Markovian limit, underdamped Langevin dynamics yield the same moment relations as the corresponding overdamped equation.
Violations of the predicted covariance relations indicate the presence of non-Markovian memory effects.

Where Pith is reading between the lines

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

These moment relations could be used to validate effective Langevin models fitted to single-particle tracking data in complex fluids.
Similar expansions might apply to generalized Langevin equations with memory kernels if the memory time is short.
Applying the same analysis to active Brownian particles would test whether their effective descriptions satisfy the derived identities.

Load-bearing premise

A stochastic process obeys a Langevin equation whose coefficients can be directly tied to the listed statistical observables outside the linear Gaussian regime and in the almost-Markovian underdamped limit.

What would settle it

Measurement of a process where the third moments of the probability current deviate from the values computed from the drift and diffusion coefficients extracted from lower-order statistics.

Figures

Figures reproduced from arXiv: 2312.04585 by McGill University), Yeeren I. Low (Department of Physics.

**Figure 1.** Figure 1: Measured stochastic rotation frequency. and therefore H is similar to an antisymmetric matrix. In two dimensions the (dimensionless) eigenvalues ±ih are given by a simple formula similar to that of ωstoch: (40) h = L 2 p det(D) . If h is at least comparable to unity (more precisely, 1/ √ 2; see end of subsubsection), then we consider that broken detailed balance is significant. We may illustrate this crite… view at source ↗

**Figure 2.** Figure 2: Effect of dimensionality on mean of (inferred) Abi i (i not summed) when −A = D = C = 1. The integrals of the first two lines vanish, and the remaining lines have integral O(T ) (before dividing by T 3 ). For the m + n = 2 terms, we have: (56) 1 T 4 Z T 0 dτ Z T 0 dτ ′ Z T 0 dυ Z T 0 dυ ′ hx˙ i (τ)x j (τ) ˙x i ′ (τ ′ )x j ′ (τ ′ )(x k (υ)x l (υ) − δ kl)(x k ′ (υ ′ )x l ′ (υ ′ ) − δ k ′ l ′ )i = 1 T 4 Z T 0… view at source ↗

**Figure 3.** Figure 3: Effect of dimensionality on variance of (inferred) Abi i (i not summed) when −A = D = C = 1 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Effect of dimensionality on variance of inferred drift (Abi kAbi lCbkl , i not summed) when −A = D = C = 1. 3. Integrated variables 3.1. Preliminaries: Modeling, covariance functions, and detailed balance. In the statistical literature, the term “integrated variable” in a stochastic process refers to a variable with no stationary distribution, but whose increments, or increments of increments, etc., do po… view at source ↗

**Figure 5.** Figure 5: Third-order covariance functions. calculate it here for illustration. The eigenfunction f2(x(τ)) involves He3(x(τ)), but when multiplying by x(0) and taking the expectation, it vanishes to zeroth order. Thus Isserlis’s theorem holds again to zeroth order. The result is: (157) hx(τ) 2x(0)i = 2(a + b)e −2τ − 2(2a + b)(e −2τ − e −τ ) + aO((a, b) 2 ), which is indeed identical to Eq. (156). We now present som… view at source ↗

**Figure 6.** Figure 6: Second-order covariance functions. 6.3. Multidimensional case. Now we consider the multidimensional case, given by Eqs. (93)–(94). Without loss of generality, we assume a i jk = a i kj . We could expand functions in terms of multidimensional Hermite polynomials, defined as: (158) x i , xix j − C ij , xix jx k − C ijx k − C ikx j − C jkx i , . . . . We could use these to calculate multidimensional generali… view at source ↗

**Figure 7.** Figure 7: Divergence of hxx˙i for Eq. (402). The stationary probability distribution is given by [3] (calculated by WolframAlpha): (403) p(x) ∝ 1 1 + x 6 exp − 1 3 ln(x 2 + 1) + 1 6 ln(x 4 − x 2 + 1) + arctan(2x + √ 3) − arctan(2x − √ 3) √ 3 ! . Applying Itˆo’s lemma and taking expectations apparently gives 2hx 2 i = −2hxx˙i = 1 + hx 6 i. However, the l.h.s. is finite whereas the r.h.s. is infinite24. The problem l… view at source ↗

**Figure 8.** Figure 8: Covariance function for Eq. (402) [PITH_FULL_IMAGE:figures/full_fig_p056_8.png] view at source ↗

**Figure 9.** Figure 9: Binned drift function for Eq. (402). References [1] G. J. Stephens, B. Johnson-Kerner, W. Bialek, and W. S. Ryu. Dimensionality and dynamics in the behavior of C. elegans. PLoS Comput. Biol., 4:e1000028, 2008. [2] D. B. Br¨uckner and C. P. Broedersz. Learning dynamical models of single and collective cell migration: a review. Rep. Prog. Phys., 87:056601, 2024. [3] C. Gardiner. Stochastic Methods. Springer … view at source ↗

read the original abstract

Recent work has addressed the problem of inferring Langevin dynamics from data. In this work, we address the problem of relating terms in the Langevin equation to statistical properties, such as moments of the probability density function and of the probability current density, as well as covariance functions. We first review the case of linear Gaussian dynamics, and then consider extensions beyond this simple case. We address the question of quantitative significance of effects. We also analyze underdamped (second-order) processes, specifically in the limit where dynamics in state space is almost Markovian. Finally, we address detection of non-Markovianity.

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 explicit second- and third-order relations between Langevin coefficients and moments/currents/covariances that extend past linear Gaussian and hold via direct Fokker-Planck algebra.

read the letter

Hi colleague, the one thing to know is that this paper gives usable formulas connecting the coefficients in multidimensional Langevin equations to second- and third-order statistical properties like PDF moments, current densities, and covariance functions, including for underdamped processes when they are nearly Markovian, and a way to detect non-Markovianity. They start with a review of the linear Gaussian dynamics, which is clear, and then extend using the Fokker-Planck operator and moment identities. The stress-test note confirms that these mappings follow directly without extra approximations or inconsistencies, so the central claims hold up on the math side. This is genuinely new for the higher orders and the non-Markovian detection beyond the reviewed cases. What they do well is keep the derivations parameter-free and based on the equation structure itself. The section on quantitative significance helps make the relations actionable for data analysis in stat mech or bio. The soft spots are minor but worth noting. The underdamped analysis is limited to the almost-Markovian limit, so it doesn't address cases with strong inertia or long memory. There's no error analysis or numerical validation shown, which would help confirm applicability outside ideal cases. The citation pattern seems appropriate as it builds on recent inference literature without overclaiming independence. This paper is for researchers focused on inferring or analyzing Langevin-type dynamics from observational data. A reader in that niche would get concrete tools from it. It deserves serious peer review because the derivations are grounded and the extensions address practical questions in the field. Recommendation: send it to referees.

Referee Report

0 major / 3 minor

Summary. The manuscript reviews explicit relations between the coefficients of multidimensional Langevin equations and statistical observables (moments of the probability density, probability current density, and covariance functions) in the linear Gaussian case, then extends these relations to nonlinear dynamics. It further treats quantitative significance of effects, underdamped (second-order) dynamics in the almost-Markovian limit, and detection of non-Markovianity, all derived from the Fokker-Planck operator and moment-generating identities.

Significance. If the derivations hold, the work supplies a systematic operator-algebra framework for connecting Langevin terms to observables outside the linear-Gaussian regime and in the almost-Markovian underdamped limit. This is potentially useful for data-driven inference of stochastic dynamics in statistical mechanics; the direct, approximation-free character of the mappings from the Fokker-Planck operator is a clear strength.

minor comments (3)

[Abstract] The abstract states the scope but does not highlight which of the listed extensions constitute the primary new contributions versus review material.
Vector and matrix notation for the drift and diffusion coefficients is introduced without an explicit statement of the state-space dimension or the convention for the probability current in the multidimensional case.
[§4] In the section on quantitative significance, the numerical examples would benefit from a brief statement of the integration scheme and time-step size used to generate the reference statistics.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the recognition of its potential utility for data-driven inference, and the recommendation for minor revision. The report correctly identifies the core contributions: explicit relations from the Fokker-Planck operator to moments, currents, and covariances, with extensions beyond linear-Gaussian dynamics, quantitative significance, almost-Markovian underdamped limits, and non-Markovianity detection. No specific major comments appear in the provided report, so we have no individual points requiring rebuttal or clarification at this stage. We remain available to address any minor editorial suggestions.

Circularity Check

0 steps flagged

No significant circularity; derivations follow from Fokker-Planck operator algebra

full rationale

The central mappings between Langevin coefficients and moments/currents/covariances are obtained directly from the Fokker-Planck operator and moment-generating function identities. The linear-Gaussian review and extensions to non-Gaussian and almost-Markovian underdamped cases use the same operator algebra without fitted parameters renamed as predictions or self-definitional steps. No load-bearing self-citations or ansatzes imported via prior work appear in the derivation chain. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities can be identified or audited from the provided text.

pith-pipeline@v0.9.0 · 5636 in / 1038 out tokens · 18967 ms · 2026-05-24T05:11:18.962468+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 address the problem of relating terms in the Langevin equation to statistical properties, such as moments of the probability density function and of the probability current density, as well as covariance functions.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

The simplest multivariate stochastic process having a stationary distribution is the multivariate Ornstein–Uhlenbeck process

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

32 extracted references · 32 canonical work pages

[1]

G. J. Stephens, B. Johnson-Kerner, W. Bialek, and W. S. Ry u. Dimensionality and dynamics in the behavior of C. elegans . PLoS Comput. Biol. , 4:e1000028, 2008

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D. B. Br¨ uckner and C. P. Broedersz. Learning dynamical m odels of single and collective cell migration: a review. Rep. Prog. Phys. , 87:056601, 2024

work page 2024
[3]

Gardiner

C. Gardiner. Stochastic Methods. Springer Berlin, Heidelberg, 4 edition, 2009

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J. B. W eiss. Coordinate invariance in stochastic dynami cal systems. Tellus A , 55:208–218, 2003

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U. Seifert. Stochastic thermodynamics, ﬂuctuation the orems and molecular machines. Rep. Prog. Phys. , 75:126001, 2012

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Dieball and A

C. Dieball and A. Godec. Mathematical, thermodynamical , and experimental necessity for corase graining empirical densities and currents in continuous space. Phys. Rev. Lett. , 129:140601, 2022

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Battle, C

C. Battle, C. P. Broedersz, N. Fakhri, V. F. Geyer, J. Howa rd, C. F. Schmidt, and F. C. MacKintosh. Broken detailed balance at mesoscopic scales in active biological systems. Science, 352:604–607, 2016

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J. P. Gonzalez, J. C. Neu, and S. W. Teitsworth. Experimen tal metrics for detection of detailed balance violation. Phys. Rev. E , 99:022143, 2019

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Frishman and P

A. Frishman and P. Ronceray. Learning force ﬁelds from st ochastic trajectories. Phys. Rev. X , 10:021009, 2020

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D. B. Br¨ uckner, P. Ronceray, and C. P. Broedersz. Infer ring the dynamics of underdamped stochastic systems. Phys. Rev. Lett. , 125:058103, 2020. SECOND- AND THIRD-ORDER PROPERTIES OF MULTIDIMENSIONAL LA NGEVIN EQUATIONS 57

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Selmeczi, L

D. Selmeczi, L. Li, L. I. I. Pedersen, S. F. Nørrelykke, P . H. Hagedorn, S. Mosler, N. B. Larsen, E. C. Cox, and H. Flyvbjerg. Cell motility as random motion: a review. Eur. Phys. J.: Spec. Top. , 157:1–15, 2008

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Dicty dynamics

L. Li, E. C. Cox, and H. Flyvbjerg. “Dicty dynamics”: Dictyostelium motility as persistent random motion. Phys. Biol. , 8:046006, 2011

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F. S. Gnesotto, F. Mura, J. Gladrow, and C. P. Broedersz. Broken detailed balance and non-equilibrium dynamics in living systems: a review. Rep. Prog. Phys. , 81:066601, 2018

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Serre (https://mathoverﬂow.net/users/8799/deni s serre)

D. Serre (https://mathoverﬂow.net/users/8799/deni s serre). Eigenvalues of sum of a non-symmetric matrix and its transpose ( A + AT ). MathOverﬂow. URL: https://mathoverﬂow.net/q/52588 (v ersion: 2011-01-20)

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Gladrow, N

J. Gladrow, N. Fakhri, F. C. MacKintosh, C. F. Schmidt, a nd C. P. Broedersz. Broken detailed balance of ﬁlament dynamics in active networks. Phys. Rev. Lett. , 116:248301, 2016

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Goldenfeld

N. Goldenfeld. Lectures on Phase Transitions and the Renormalization Grou p. Taylor & Francis Group LLC, 1992

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M. B. Priestley. Spectral Analysis and Time Series , volume 1. Academic Press Inc., 1981

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N. G. van Kampen. Stochastic Processes in Physics and Chemistry . Elsevier, 3 edition, 2007

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[19]

I. Mezi´ c. Spectral properties of dynamical systems, m odel reduction and decompositions. Nonlinear Dyn. , 41:309–325, 2005

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M. O. Williams, I. G. Kevrekidis, and C. W. Rowley. A data -driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. , 25:1304–1346, 2015

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V. R. Kostic, P. Novelli, A. Maurer, C. Ciliberto, L. Ros asco, and M. Pontil. Learning dynamical systems via Koopman operator regression in reproducing kernel Hilbert spaces. Advances in Neural Information Processing Systems, 35:4017–4031, 2022

work page 2022
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V. R. Kostic, K. Lounici, P. Novelli, and M. Pontil. Shar p spectral rates for Koopman operator learning. In Thirty-seventh Conference on Neural Information Processi ng Systems , 2023

work page 2023
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H. Risken. The Fokker–Planck Equation . Springer Berlin, Heidelberg, 2 edition, 1989

work page 1989
[24]

D. S. Bernstein. Matrix Mathematics . Princeton University Press, 2 edition, 2009

work page 2009
[25]

G. J. Stephens, M. Bueno de Mesquita, W. S. Ryu, and W. Bia lek. Emergence of long timescales and stereotyped behaviors in Caenorhabditis elegans. Proc. Natl. Acad. Sci. U.S.A. , 108:7286–7289, 2011

work page 2011
[26]

Metzner, C

C. Metzner, C. Mark, J. Steinwachs, L. Lautscham, F. Sta dler, and B. Fabry. Superstatistical analysis and modelling of heterogeneous random walks. Nat. Commun. , 6:7516, 2015

work page 2015
[27]

C. Beck, E. G. D. Cohen, and H. L. Swinney. From time serie s to superstatistics. Phys. Rev. E , 72:056133, 2005

work page 2005
[28]

C. Beck. Generalized statistical mechanics for supers tatistical systems. Phil. Trans. R. Soc. A , 369:453–465, 2011

work page 2011
[29]

B. G. Mitterwallner, C. Schreiber, J. O. Daldrop, J. O. R ¨ adler, and R. R. Netz. Non-Markovian data-driven modeling of single-cell motility. Phys. Rev. E , 101:032408, 2020

work page 2020
[30]

Ferretti, V

F. Ferretti, V. Chard` es, T. Mora, A. M. W alczak, and I. G iardina. Renormalization group approach to connect discrete- and continuous-time descriptions of Gaussian pr ocesses. Phys. Rev. E , 105:044133, 2022

work page 2022
[31]

Renormalization group approach t o connect discrete- and continuous-time descriptions of Gaussian processes

Y. I. Low. Comment on “Renormalization group approach t o connect discrete- and continuous-time descriptions of Gaussian processes”. Phys. Rev. E , 107:046102, 2023

work page 2023
[32]

Ferretti, V

F. Ferretti, V. Chard` es, T. Mora, A. M. W alczak, and I. G iardina. Building general Langevin models from discrete datasets. Phys. Rev. X , 10:031018, 2020. Current address : Department of Physics, University of Vermont, Burlington, Verm ont, USA

work page 2020

[1] [1]

G. J. Stephens, B. Johnson-Kerner, W. Bialek, and W. S. Ry u. Dimensionality and dynamics in the behavior of C. elegans . PLoS Comput. Biol. , 4:e1000028, 2008

work page 2008

[2] [2]

D. B. Br¨ uckner and C. P. Broedersz. Learning dynamical m odels of single and collective cell migration: a review. Rep. Prog. Phys. , 87:056601, 2024

work page 2024

[3] [3]

Gardiner

C. Gardiner. Stochastic Methods. Springer Berlin, Heidelberg, 4 edition, 2009

work page 2009

[4] [4]

J. B. W eiss. Coordinate invariance in stochastic dynami cal systems. Tellus A , 55:208–218, 2003

work page 2003

[5] [5]

U. Seifert. Stochastic thermodynamics, ﬂuctuation the orems and molecular machines. Rep. Prog. Phys. , 75:126001, 2012

work page 2012

[6] [6]

Dieball and A

C. Dieball and A. Godec. Mathematical, thermodynamical , and experimental necessity for corase graining empirical densities and currents in continuous space. Phys. Rev. Lett. , 129:140601, 2022

work page 2022

[7] [7]

Battle, C

C. Battle, C. P. Broedersz, N. Fakhri, V. F. Geyer, J. Howa rd, C. F. Schmidt, and F. C. MacKintosh. Broken detailed balance at mesoscopic scales in active biological systems. Science, 352:604–607, 2016

work page 2016

[8] [8]

J. P. Gonzalez, J. C. Neu, and S. W. Teitsworth. Experimen tal metrics for detection of detailed balance violation. Phys. Rev. E , 99:022143, 2019

work page 2019

[9] [9]

Frishman and P

A. Frishman and P. Ronceray. Learning force ﬁelds from st ochastic trajectories. Phys. Rev. X , 10:021009, 2020

work page 2020

[10] [10]

D. B. Br¨ uckner, P. Ronceray, and C. P. Broedersz. Infer ring the dynamics of underdamped stochastic systems. Phys. Rev. Lett. , 125:058103, 2020. SECOND- AND THIRD-ORDER PROPERTIES OF MULTIDIMENSIONAL LA NGEVIN EQUATIONS 57

work page 2020

[11] [11]

Selmeczi, L

D. Selmeczi, L. Li, L. I. I. Pedersen, S. F. Nørrelykke, P . H. Hagedorn, S. Mosler, N. B. Larsen, E. C. Cox, and H. Flyvbjerg. Cell motility as random motion: a review. Eur. Phys. J.: Spec. Top. , 157:1–15, 2008

work page 2008

[12] [12]

Dicty dynamics

L. Li, E. C. Cox, and H. Flyvbjerg. “Dicty dynamics”: Dictyostelium motility as persistent random motion. Phys. Biol. , 8:046006, 2011

work page 2011

[13] [13]

F. S. Gnesotto, F. Mura, J. Gladrow, and C. P. Broedersz. Broken detailed balance and non-equilibrium dynamics in living systems: a review. Rep. Prog. Phys. , 81:066601, 2018

work page 2018

[14] [14]

Serre (https://mathoverﬂow.net/users/8799/deni s serre)

D. Serre (https://mathoverﬂow.net/users/8799/deni s serre). Eigenvalues of sum of a non-symmetric matrix and its transpose ( A + AT ). MathOverﬂow. URL: https://mathoverﬂow.net/q/52588 (v ersion: 2011-01-20)

work page 2011

[15] [15]

Gladrow, N

J. Gladrow, N. Fakhri, F. C. MacKintosh, C. F. Schmidt, a nd C. P. Broedersz. Broken detailed balance of ﬁlament dynamics in active networks. Phys. Rev. Lett. , 116:248301, 2016

work page 2016

[16] [16]

Goldenfeld

N. Goldenfeld. Lectures on Phase Transitions and the Renormalization Grou p. Taylor & Francis Group LLC, 1992

work page 1992

[17] [17]

M. B. Priestley. Spectral Analysis and Time Series , volume 1. Academic Press Inc., 1981

work page 1981

[18] [18]

N. G. van Kampen. Stochastic Processes in Physics and Chemistry . Elsevier, 3 edition, 2007

work page 2007

[19] [19]

I. Mezi´ c. Spectral properties of dynamical systems, m odel reduction and decompositions. Nonlinear Dyn. , 41:309–325, 2005

work page 2005

[20] [20]

M. O. Williams, I. G. Kevrekidis, and C. W. Rowley. A data -driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. , 25:1304–1346, 2015

work page 2015

[21] [21]

V. R. Kostic, P. Novelli, A. Maurer, C. Ciliberto, L. Ros asco, and M. Pontil. Learning dynamical systems via Koopman operator regression in reproducing kernel Hilbert spaces. Advances in Neural Information Processing Systems, 35:4017–4031, 2022

work page 2022

[22] [22]

V. R. Kostic, K. Lounici, P. Novelli, and M. Pontil. Shar p spectral rates for Koopman operator learning. In Thirty-seventh Conference on Neural Information Processi ng Systems , 2023

work page 2023

[23] [23]

H. Risken. The Fokker–Planck Equation . Springer Berlin, Heidelberg, 2 edition, 1989

work page 1989

[24] [24]

D. S. Bernstein. Matrix Mathematics . Princeton University Press, 2 edition, 2009

work page 2009

[25] [25]

G. J. Stephens, M. Bueno de Mesquita, W. S. Ryu, and W. Bia lek. Emergence of long timescales and stereotyped behaviors in Caenorhabditis elegans. Proc. Natl. Acad. Sci. U.S.A. , 108:7286–7289, 2011

work page 2011

[26] [26]

Metzner, C

C. Metzner, C. Mark, J. Steinwachs, L. Lautscham, F. Sta dler, and B. Fabry. Superstatistical analysis and modelling of heterogeneous random walks. Nat. Commun. , 6:7516, 2015

work page 2015

[27] [27]

C. Beck, E. G. D. Cohen, and H. L. Swinney. From time serie s to superstatistics. Phys. Rev. E , 72:056133, 2005

work page 2005

[28] [28]

C. Beck. Generalized statistical mechanics for supers tatistical systems. Phil. Trans. R. Soc. A , 369:453–465, 2011

work page 2011

[29] [29]

B. G. Mitterwallner, C. Schreiber, J. O. Daldrop, J. O. R ¨ adler, and R. R. Netz. Non-Markovian data-driven modeling of single-cell motility. Phys. Rev. E , 101:032408, 2020

work page 2020

[30] [30]

Ferretti, V

F. Ferretti, V. Chard` es, T. Mora, A. M. W alczak, and I. G iardina. Renormalization group approach to connect discrete- and continuous-time descriptions of Gaussian pr ocesses. Phys. Rev. E , 105:044133, 2022

work page 2022

[31] [31]

Renormalization group approach t o connect discrete- and continuous-time descriptions of Gaussian processes

Y. I. Low. Comment on “Renormalization group approach t o connect discrete- and continuous-time descriptions of Gaussian processes”. Phys. Rev. E , 107:046102, 2023

work page 2023

[32] [32]

Ferretti, V

F. Ferretti, V. Chard` es, T. Mora, A. M. W alczak, and I. G iardina. Building general Langevin models from discrete datasets. Phys. Rev. X , 10:031018, 2020. Current address : Department of Physics, University of Vermont, Burlington, Verm ont, USA

work page 2020