Pith. sign in

REVIEW 2 major objections 6 minor 60 references

Data-driven approximation of transfer operators for mean-field stochastic differential equations

T0 review · 2 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read EDMD applied to the decoupled McKean–Vlasov SDE converges almost surely to the exact projected Koopman operator.

desk verdict Solid EDMD convergence theory for a decoupled McKean-Vlasov SDE, but the paper's advertised metastability claims rest on an unproven leap from a time-inhomogeneous finite-time operator to the nonlinear mean-field system. read the letter →

arxiv 2509.09891 v1 pith:NSWZ7KDM submitted 2025-09-11 math.DS cs.NAmath.NAmath.PRstat.ML

Eirini Ioannou , Stefan Klus , Gonçalo dos Reis This is my paper · ORCID
classification math.DScs.NAmath.NAmath.PRstat.ML MSC 37M2560H1065C30
keywords transferoperatorsKoopmanoperatorPerron–FrobeniusMcKean–VlasovSDEmean-fielddynamicsEDMDGalerkinprojectionmetastability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that transfer operator theory—the Koopman and Perron–Frobenius operators that expose global dynamical behaviour—can be extended to mean-field (McKean–Vlasov) stochastic differential equations, the limiting dynamics of large interacting particle systems. The key move is to work with a 'decoupled' McKean–Vlasov SDE in which the law of the process is a fixed time-dependent parameter; this restores the linearity and Markov property that transfer operators need. The authors prove that the data-driven EDMD estimator of the Galerkin-projected Koopman operator converges almost surely to the exact projected operator as the number of trajectories goes to infinity and the time step goes to zero. This provides the theoretical foundation for using spectral methods to detect metastable sets, slow timescales, and invariant distributions in mean-field systems, with numerical demonstrations on the Cormier and Kuramoto models.

What carries the argument

The decoupled McKean–Vlasov SDE (equation 3) is the central object: it is an ordinary Itô diffusion whose coefficients depend on the fixed law $\mu_t$ of the original mean-field limit. This restores the Markov property and makes the Koopman operator $K_T f(x) = E[f(X^{x,\mu}_T)]$ linear and contractive. The finite-dimensional approximation is built on the Galerkin identity $K_N^T = C_N G_N^{-1}$, where $C_N$ and $G_N$ are the structure and Gram matrices of the dictionary basis; EDMD replaces these exact moments by Monte Carlo averages over independent trajectories of the decoupled scheme (equations 16–17). The almost-sure convergence is carried by a strong law of large numbers for i.i.d. samples plus Lips

What would settle it

Take a mean-field SDE with a time-periodic law $\mu_t$ (e.g., a forced Kuramoto model whose center of mass oscillates), simulate the decoupled scheme for $M=10^6$, $h=0.001$, and compute the second eigenvalue $\lambda_2$ of the EDMD matrix with a fine dictionary. If $\lambda_2 \approx 1$ while direct particle simulations show all particles mixing between the candidate sets identified by the corresponding eigenfunction at the same time scale, then the claim that eigenvalues near one detect metastable sets would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that a data-driven approximation of transfer operators for mean-field SDEs is legitimate. By decoupling the law from the dynamics—replacing the McKean–Vlasov SDE with the standard SDE $dX_t = b(t,X_t,\mu_t)dt + \sigma(t,X_t,\mu_t)dW_t$ where $\mu_t$ is the original law—the Koopman and Perron–Frobenius operators become well-defined linear operators between function spaces. The paper proves (Theorem 4.7) that the EDMD matrix $\hat{K}_{N,M} = \hat{C}_{N,M}\hat{G}_{N,M}^{-1}$ converges almost surely to $C_N G_N^{-1}$ as the sample size $M \to \infty$ and, subsequently, as the discretization step $h \to 0$ along with the error in the decoupling measure. This means the eigenvalues and eigenvectors of the computed m

Load-bearing premise

The spectral interpretation—that eigenvalues close to one indicate slow timescales and metastable sets—assumes the decoupled process behaves like an autonomous Markov process; but when the law $\mu_t$ is time-dependent, the family of finite-time Koopman operators is not a semigroup, and no theorem in the paper connects the spectrum of a single operator to the metastability of the nonlinear McKean–Vlasov equation.

Editorial extensions

If this is right

  • EDMD can be applied to McKean–Vlasov SDEs with rigorous convergence guarantees, so spectral quantities (eigenvalues, eigenfunctions) reported from data are provably close to the projected Koopman operator.
  • Metastable sets and transition timescales of the decoupled process—and, heuristically, of the mean-field system—can be read off from the dominant eigenvalues of the estimated matrix.
  • The Perron–Frobenius operator is approximated by the transpose-based estimator \hat P_{N,M}^T = \hat C_{N,M}^T \hat G_{N,M}^{-1}, so invariant densities and eigenfunctions are accessible with the same data.
  • If particle-system data are used instead of decoupled data, only L2 convergence holds; the decoupled scheme is the one that yields almost-sure convergence and clean eigenvalue estimates.
  • The method identifies the known metastable hemispheres in the sphere Kuramoto model and the two metastable intervals in the circle Kuramoto model, matching analytical predictions.

Reading between the lines

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

  • The convergence theorem suggests a practical recipe: estimate the law µ via a short particle simulation, fix it, then run many independent decoupled trajectories; the cost separates into law estimation and EDMD estimation, which may allow adaptive refinement of each.
  • The spectral interpretation is the least secure part: since µ_t varies in time, the Koopman family does not form a semigroup, so a single matrix's spectrum may not correspond to physically meaningful timescales for time-dependent mean-field dynamics. A direct test would be to compare EDMD eigenvalues with transition rates computed by long-time particle simulations for a non-stationary µ_t.
  • The framework naturally extends to generator-based EDMD (gEDMD) on the decoupled SDE, which would eliminate the arbitrary lag time T and might yield sharper spectra.
  • Because the almost-sure result depends only on the vanishing of sup_k W2(µ_{t_k}, \hat µ_{t_k}), more efficient sampling of the law (multilevel, quasi-Monte Carlo) could be plugged in without changing the main proof structure.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper develops transfer-operator methodology for McKean–Vlasov SDEs. Since the McKean–Vlasov equation is nonlinear, the authors work with the decoupled SDE (3), in which the law µ_t is treated as an externally prescribed time-dependent coefficient. They define Koopman and Perron–Frobenius operators for this decoupled process, form Galerkin projections onto a finite dictionary with respect to the initial law µ0, and estimate the projected matrices by Monte Carlo EDMD. The central theoretical result (Theorem 4.7) asserts almost-sure convergence of the data-driven EDMD matrix to (C_N + E(h))G_N^{-1} as the number of samples M tends to infinity, and then to the exact projected Koopman matrix C_N G_N^{-1} as the time step h tends to zero. The paper also presents numerical experiments for the Cormier model, the Kuramoto model on the circle, and a Kuramoto model on the sphere, interpreting eigenvalues close to one as evidence of metastability.

Significance. If the results are taken as stated, the paper provides a rigorous convergence guarantee for EDMD in a mean-field SDE setting, extending a well-established data-driven framework to a class of nonlinear, measure-dependent dynamics. The proof strategy is largely sound: the Gram and structure matrices are Monte Carlo estimates of well-defined expectations, and no parameter is fit to reproduce a target spectrum. The numerical examples illustrate the method on nontrivial benchmark models. However, the advertised spectral interpretation in terms of metastability and slow timescales is not established for the time-inhomogeneous decoupled operator, and the definition of the transfer operators contains a measure-theoretic inconsistency. These issues concern the main interpretive claim of the paper, so the contribution, while promising, is not yet fully supported.

major comments (2)
  1. [§3.1, Eq. (8); §3.2, Lemma 3.1] The transition density p is introduced in §2.2 as a spatial probability density (see Eq. (4), where q(µ0,0,t,z)=∫ p(µ0,0,t,x,z) µ0(dx)). But Eq. (8) defines K_T f(x)=∫ p(µ0,0,T,x,y) f(y) µ0(dy), and the proof of Lemma 3.1 uses ∫ p(µ0,0,T,x,y) µ0(dy)=1. These identities are incompatible with p being a density with respect to Lebesgue measure unless µ0 is Lebesgue measure. Thus the definitions of K_T and P_T as written are not well-defined for a general initial law µ0. The Monte Carlo target C_N in Eq. (13) uses the correct expectation E[ψ_i(X_T^{ξ,µ})ψ_j(ξ)], so the convergence theorem can be salvaged, but the operator definitions and the duality statement must be rewritten consistently (for example, by using Lebesgue measure as the reference and deriving the appropriate duality, or by explicitly defining p as a density with respect to µ0 and adjusting Eq. (4)).
  2. [§3.5 and §5] The spectral/metastability interpretation is an unproven extrapolation. For the decoupled SDE (3), the coefficients depend on t through µ_t, so the family {K_T}_{T>0} is not a semigroup, and K_T itself depends on the initial law µ0. The standard results cited (e.g., [27,50]) concern autonomous Markov semigroups. No theorem in this paper connects the eigenvalues of a single finite-time K_T to slow timescales or metastable sets of the nonlinear McKean–Vlasov system. The experiments in §5.2 start from uniform initial data rather than an invariant measure, so the reported second eigenvalue could reflect transient relaxation rather than a genuine metastable transition. The central advertised claim—detection of metastable sets—therefore requires either a proof under additional assumptions or a substantial caveat limiting the claim to the auxiliary decoupled process.
minor comments (6)
  1. [§2.1, Eq. (2)] In Eq. (2), the notation 'X_t = ξ' should read 'X_0 = ξ'.
  2. [§2.2, Eq. (3)] The notation 'µ=Law(X_t)' in Eq. (3) is ambiguous and appears circular. The superscript µ is used as the fixed law flow of the McKean–Vlasov SDE (2), not the law of the decoupled process. This should be clarified.
  3. [§3.4] The statement that 'both Koopman and Perron–Frobenius operators can be extended or restricted to be defined on L2(X,µ0)' is asserted without justification. For a non-stationary process and arbitrary f∈L2(µ0), the expectation E[f(X_T^x)] need not be finite or well-defined µ0-a.e. For the bounded dictionary used here this is harmless, but the extension claim should be formulated more carefully or omitted.
  4. [§4.2, Lemma 4.3] The L2(Ω) convergence of \widehat G_N^{-1} to G_N^{-1} is asserted in the proof of Theorem 4.6 but not proved in Lemma 4.3. It follows from continuity of inversion on the set of invertible matrices, but this step should be stated explicitly.
  5. [§5.2] The sentence 'Since all except the first two eigenvalues are small, there is only one form of metastability' is imprecise: the number of metastable sets is not determined solely by the number of eigenvalues close to one without additional assumptions about the operator and the system.
  6. [Throughout] There are several typos: 'Mckean' in the first paragraph of Section 4; 'X_t = ξ' in Eq. (2); and in Assumption 3.3(a) the bound involving γ_N appears to contain a typo (it should likely be |(ψ_1(x),...,ψ_N(x))|^2 < γ_N or a matching definition).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the EDMD convergence proof is self-contained and not reduced to its inputs; the main caveat is an unproven spectral extrapolation, not a circular step.

full rationale

The central derivation chain is not circular. The data-driven matrices (16)-(17) are Monte Carlo estimates of well-defined expectations: \hat G_N^M estimates G_N = E[\psi_i(\xi)\psi_j(\xi)] with \xi~\mu_0, and \hat C_N^M estimates C_N = E[\psi_i(X^{\xi,\mu}_T)\psi_j(\xi)], where X^{\xi,\mu} solves the decoupled SDE (3) with the true McKean-Vlasov law \mu_t. The target C_N G_N^{-1} is the Galerkin projection of the finite-time Koopman operator (8); it depends on \mu only through the SDE coefficients, not through the fitted output. Theorem 4.7 decomposes the error into a Monte Carlo term and a deterministic bias E(h) coming from the estimated law \hat\mu and time step h, and shows both vanish in the successive limits M->infty and h->0 using the strong law of large numbers and external Wasserstein-convergence results (Theorem 2.3). No parameter is fit to reproduce a spectrum, so no fitted input is renamed as a prediction. The paper leans on the self-cited preprint [49] for parts of the EDMD framework and for the eigenvalue-convergence corollary, and [50] (also author-overlapping) for the timescale/metastability vocabulary. However, the Gram-matrix and structure-matrix convergence proofs are reproduced locally, and the eigenvalue-continuity step rests on standard matrix perturbation theory [52,53]; the self-citations are therefore not load-bearing in the sense of forcing the conclusion. The substantive caveat is in Section 3.5: the spectrum of the single finite-time operator K_T of the time-inhomogeneous decoupled process is asserted, citing [50], to indicate metastability/timescales of the mean-field system, but no semigroup or metastability theorem is proved for this non-autonomous operator. That is an unproven extrapolation and a correctness risk, not a circular reduction, because convergence to K_T is not equivalent to the spectral interpretation. Overall circularity score: 1.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central convergence theorem is derived from standard martingale and strong law arguments under four assumptions. The only axiom specific to this paper's interpretive claim is the identification of K_T eigenvalues with metastable timescales in Section 3.5, imported from autonomous Markov process theory without justification for time-varying mu_t. The numerical settings (basis, lag, step) are user choices that affect the reported spectra but are not part of the proof.

free parameters (3)
  • Dictionary basis set = indicator functions on intervals, monomials up to order 7, 200 Voronoi cells across examples
    The Galerkin projection and the resulting eigenvalue estimates depend on the chosen basis; no automated or data-driven basis selection is used.
  • Lag time T = 0.5 for Cormier and sphere, 1 for circle in the experiments
    Eigenvalues of K_T scale with T; T is chosen by hand, and the theory requires T fixed.
  • Time step h and IPS particle count = h=0.1 with 500000 particles for Cormier, h=0.01 with 5000 particles for Kuramoto models
    The numerical scheme discretization and the law estimation introduce bias that vanishes only in the limit; the concrete values are hand-picked.
assumptions (4)
  • domain assumption Assumption 2.2: drift and diffusion are uniformly Lipschitz in space and measure, and 1/2-Holder in time, ensuring well-posedness of the McKean-Vlasov and decoupled SDEs.
    Invoked throughout Sections 2-4, for example in Theorem 2.3 and Lemma 4.5.
  • domain assumption Assumptions 3.2 and 3.3: basis functions are linearly independent, bounded, Lipschitz, and K_T psi_n continuous almost everywhere.
    Needed for invertibility of the Gram matrix and for Monte Carlo convergence in Lemmas 4.3 and 4.4.
  • domain assumption Assumption 4.1: data pairs are i.i.d. from a fixed decoupling measure hat_mu, and sup_k W2(mu_{t_k}, hat_mu_{t_k}) goes to 0 as h goes to 0.
    This is the data regime that permits the strong law of large numbers in Lemma 4.5; it fixes hat_mu to avoid dependence among samples.
  • ad hoc to paper Eigenvalues of K_T close to one correspond to slow timescales and metastable sets, citing [50].
    Section 3.5 applies a result for autonomous Markov processes to the time-inhomogeneous decoupled process without proving the semigroup property or stationarity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Data-driven approximation of transfer operators for mean-field stochastic differential equations." pith.science (2026). https://pith.science/paper/NSWZ7KDM

@misc{pith2026250909891,
  author       = {Pith},
  title        = {Pith review of: Data-driven approximation of transfer operators for mean-field stochastic differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSWZ7KDM}},
  note         = {Machine review of arXiv:2509.09891}
}
read the original abstract

Mean-field stochastic differential equations, also called McKean--Vlasov equations, are the limiting equations of interacting particle systems with fully symmetric interaction potential. Such systems play an important role in a variety of fields ranging from biology and physics to sociology and economics. Global information about the behavior of complex dynamical systems can be obtained by analyzing the eigenvalues and eigenfunctions of associated transfer operators such as the Perron--Frobenius operator and the Koopman operator. In this paper, we extend transfer operator theory to McKean--Vlasov equations and show how extended dynamic mode decomposition and the Galerkin projection methodology can be used to compute finite-dimensional approximations of these operators, which allows us to compute spectral properties and thus to identify slowly evolving spatiotemporal patterns or to detect metastable sets. The results will be illustrated with the aid of several guiding examples and benchmark problems including the Cormier model, the Kuramoto model, and a three-dimensional generalization of the Kuramoto model.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

60 extracted references · 19 canonical work pages

  1. [49]

    Llamazares-Elias, S

    L. Llamazares-Elias, S. Llamazares-Elias, J. Latz, and S. Klus. Data-driven approximation of Koopman operators and generators: Convergence rates and error bounds.arXiv preprint arXiv:2405.00539, 2024.doi:10.48550/arXiv.2405.00539

    arXiv doi:10.48550/arxiv.2405.00539 2024

  2. [1]

    Giacomin, S

    G. Giacomin, S. Olla, E. Saada, H. Spohn, and G. Stoltz.Stochastic Dynamics Out of Equilibrium. Springer Proceedings in Mathematics and Statistics. Springer Cham, 2017. doi:10.1007/978-3-030-15096-9

    work page doi:10.1007/978-3-030-15096-9 2017

  3. [2]

    Baladron, D

    J. Baladron, D. Fasoli, O. Faugeras, and J. Touboul. Mean field description of and propa- gation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and FitzHugh- Nagumo neurons, 2012.doi:10.1186/2190-8567-2-10

    work page doi:10.1186/2190-8567-2-10 2012

  4. [3]

    Primi, A

    I. Primi, A. Stevens, and J. J. L. Velazquez. Mass-selection in alignment models with non- 25 deterministic effects.Communications in Partial Differential Equations, 34(5):419–456, 2009.doi:10.1080/03605300902797171

    work page doi:10.1080/03605300902797171 2009

  5. [4]

    Peruani, A

    F. Peruani, A. Deutsch, and M. Bär. A mean-field theory for self-propelled particles interacting by velocity alignment mechanisms.European Physical Journal Special topics, 157:111–122, 2008.doi:10.1140/epjst/e2008-00634-x

    work page doi:10.1140/epjst/e2008-00634-x 2008

  6. [5]

    Fiorelli, M

    E. Fiorelli, M. Mueller, I. Lesanovsky, and F. Carollo. Mean-field dynamics of open quan- tum systems with collective operator-valued rates: validity and application.New Journal of Physics, 25(8):083010, 2023.doi:10.1088/1367-2630/ace470

    work page doi:10.1088/1367-2630/ace470 2023

  7. [6]

    Chazelle, Q

    B. Chazelle, Q. Jiu, Q. Li, and C. Wang. Well-posedness of the limiting equation of a noisy consensus model in opinion dynamics.Journal of Differential Equations, 263(1):365–397, 2017.doi:10.1016/j.jde.2017.02.036

    doi:10.1016/j.jde.2017.02.036 2017

  8. [7]

    Garnier, G

    J. Garnier, G. Papanicolaou, and T.-W. Yang. Consensus convergence with stochas- tic effects.Vietnam Journal of Mathematics, 45(1):51–75, 2017.doi:10.1007/ s10013-016-0190-2

    2017

Show all 60 references
  1. [8]

    B. D. Goddard, B. Gooding, H. Short, and G. A. Pavliotis. Noisy bounded confidence models for opinion dynamics: the effect of boundary conditions on phase transitions.IMA Journal of Applied Mathematics, 87(1):80–110, 2022.doi:10.1093/imamat/hxab044

    2022 doi

  2. [9]

    Helfmann, N

    L. Helfmann, N. Djurdjevac Conrad, P. Lorenz-Spreen, and C. Schütte. Modelling opin- ion dynamics under the impact of influencer and media strategies.Scientific reports, 13(1):19375, 2023.doi:10.1038/s41598-023-46187-9

    2023 doi

  3. [10]

    Degond, J.-G

    P. Degond, J.-G. Liu, and C. Ringhofer. Large-scale dynamics of mean-field games driven by local Nash equilibria.Journal of Nonlinear Science, 24(1):93–115, 2014.doi:10.1007/ s00332-013-9185-2

    2014

  4. [11]

    Fouque and L.-H

    J.-P. Fouque and L.-H. Sun.Handbook on Systemic Risk, chapter Systemic Risk Illustrated, pages 444–452. Systemic risk and mathematical finance. Cambridge University Press, 2013

    2013

  5. [12]

    Garnier, G

    J. Garnier, G. Papanicolaou, and T.-W. Yang. Large deviations for a mean field model of systemic risk.SIAM Journal on Financial Mathematics, 4(1):151–184, 2013.doi: 10.1137/12087387X

    2013 doi

  6. [13]

    Lasry and P.-L

    J.-M. Lasry and P.-L. Lions.Mean Field Games, volume 9, chapter I: The stationary case, pages 619–625. Elsevier, 2006

    2006

  7. [14]

    T. D. Frank.Nonlinear Fokker–Planck Equations Fundamentals and Applications. Num- ber 1 in Springer Series in Synergetics. Springer Berlin, Heidelberg, Heidelberg, 2005. doi:10.1007/b137680

    2005 doi

  8. [15]

    Naldi, L

    G. Naldi, L. Pareschi, and G. Toscani.Mathematical modeling of collective behavior in socio-economic and life sciences. Number 1 in Modeling and Simulation in Sci- ence, Engineering and Technology. Birkhäuser Boston, Ltd., Boston, MA, 2010.doi: 10.1007/978-0-8176-4946-3

    2010 doi

  9. [16]

    D. Lacker. Mean field games and interacting particle systems, lecture notes, 2018. Columbia University. URL:https://www.columbia.edu/~dl3133/MFGSpring2018.pdf. 26

    2018

  10. [17]

    B. Koopman. Hamiltonian systems and transformations in Hilbert space.Proceedings of the National Academy of Sciences, 17(5):315, 1931

    1931

  11. [18]

    Lasota and M

    A. Lasota and M. C. Mackey.Chaos, fractals, and noise: Stochastic aspects of dynamics, volume 97 ofApplied Mathematical Sciences. Springer, 2nd edition, 1994

    1994

  12. [19]

    Dellnitz and O

    M. Dellnitz and O. Junge. On the approximation of complicated dynamical behavior. SIAM Journal on Numerical Analysis, 36(2):491–515, 1999

    1999

  13. [20]

    I. Mezić. Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynamics, 41(1):309–325, 2005.doi:10.1007/s11071-005-2824-x

    2005 doi

  14. [21]

    S. Klus, P. Koltai, and C. Schütte. On the numerical approximation of the Perron– Frobenius and Koopman operator.Journal of Computational Dynamics, 3(1):51–79, 2016. doi:10.3934/jcd.2016003

    2016 doi

  15. [22]

    M. O. Williams, I. G. Kevrekidis, and C. W. Rowley. A data-driven approximation of the Koopman operator: extending dynamic mode decomposition.Journal of Nonlinear Science, 25:1307–1346, 2015.doi:10.1007/s00332-015-9258-5

    2015 doi

  16. [23]

    M. O. Williams, C. W. Rowley, and I. G. Kevrekidis. A kernel-based method for data- driven Koopman spectral analysis.Journal of Computational Dynamics, 2(2):247–265, 2015

    2015

  17. [24]

    S. Klus, I. Schuster, and K. Muandet. Eigendecompositions of transfer operators in reproducing kernel Hilbert spaces.Journal of Nonlinear Science, 30:283–315, 2019. doi:10.1007/s00332-019-09574-z

    2019 doi

  18. [25]

    S. Klus, F. Nüske, S. Peitz, J.-H. Niemann, C. Clementi, and C. Schütte. Data-driven approximation of the Koopman generator: Model reduction, system identification, and control.Physica D: Nonlinear Phenomena, 406:132416, 2020.doi:10.1016/j.physd. 2020.132416

    2020

  19. [26]

    Mardt, L

    A. Mardt, L. Pasquali, H. Wu, and F. Noé. VAMPnets for deep learning of molecular kinetics.Nature Communications, 9, 2018.doi:10.1038/s41467-017-02388-1

    2018 doi

  20. [27]

    Schütte and M

    C. Schütte and M. Sarich.Metastability and Markov State Models in Molecular Dynam- ics: Modeling, Analysis, Algorithmic Approaches. Number 24 in Courant Lecture Notes. American Mathematical Society, 2013

    2013

  21. [28]

    Noé and F

    F. Noé and F. Nüske. A variational approach to modeling slow processes in stochastic dynamical systems.Multiscale Modeling & Simulation, 11(2):635–655, 2013

    2013

  22. [29]

    Martzel and C

    N. Martzel and C. Aslangul. Mean-field treatment of the many-body Fokker–Planck equation.Journal of Physics A: Mathematical and General, 34(50):11225–11240, 2001. doi:10.1088/0305-4470/34/50/305

    2001 doi

  23. [30]

    J. A. Carrillo, R. S. Gvalani, G. A. Pavliotis, and A. Schlichting. Long-time behaviour and phase transitions for the McKean-Vlasov equation on the torus.Archive for Rational Mechanics and Analysis, 235(1):635–690, 2020.doi:10.1007/s00205-019-01430-4. 27

    2020 doi

  24. [31]

    Bertoli, B

    B. Bertoli, B. D. Goddard, and G. A. Pavliotis. Stability of stationary states for mean field models with multichromatic interaction potentials.IMA Journal of Applied Mathematics, 89(5):833–859, 2025.doi:10.1093/imamat/hxaf001

    2025 doi

  25. [32]

    Bicego, D

    S. Bicego, D. Kalise, and G. A. Pavliotis. Computation and control of unstable steady states for mean field multiagent systems.Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, 481:20240476, 2025.doi: 10.1098/rspa.2024.0476

    2025

  26. [33]

    Q. Cormier. On the stability of the invariant probability measures of McKean–Vlasov equations.arXiv preprint arXiv:2201.11612, 2024.doi:arXiv:2201.11612

    2024 arXiv

  27. [34]

    G. A. Pavliotis and A. Zanoni. Eigenfunction martingale estimators for interacting par- ticle systems and their mean field limit.SIAM Journal on Applied Dynamical Systems, 21(4):2338–2370, 2022.doi:10.1137/21M1464348

    2022 doi

  28. [35]

    Della Maestra and M

    L. Della Maestra and M. Hoffmann. Nonparametric estimation for interacting particle systems: McKean–Vlasov models.Probability theory and related fields, 182(1):551–613, 2022.doi:10.1007/s00440-021-01044-6

    2022 doi

  29. [36]

    Sharrock, N

    L. Sharrock, N. Kantas, P. Parpas, and G. A. Pavliotis. Online parameter estimation for the McKean–Vlasov stochastic differential equation.Stochastic Processes and their Applications, 162:481–546, 2023.doi:10.1016/j.spa.2023.05.002

    2023 doi

  30. [37]

    Della Maestra and M

    L. Della Maestra and M. Hoffmann. The LAN property for Mckean–Vlasov models in a mean-field regime.Stochastic Processes and their Applications, 155:109–146, 2023.doi: 10.1016/j.spa.2022.10.002

    2023 doi

  31. [38]

    Crisan and E

    D. Crisan and E. McMurray. Smoothing properties of McKean–Vlasov SDEs.Probability Theory and Related Fields, 171:97–148, 2018.doi:10.1007/s00440-017-0774-0

    2018 doi

  32. [39]

    Buckdahn, J

    R. Buckdahn, J. Li, S. Peng, and C. Rainer. Mean-field stochastic differential equations and associated PDEs.The Annals of Probability, 45(2):824–878, 2017.doi:10.1214/ 15-AOP1076

    2017

  33. [40]

    Chen and G

    X. Chen and G. dos Reis. Euler simulation of interacting particle systems and McKean– Vlasov SDEs with fully super-linear growth drifts in space and interaction.IMA Journal of Numerical Analysis, 44(2):751–796, 2024.doi:10.1093/imanum/drad022

    2024 doi

  34. [41]

    Belomestny and J

    D. Belomestny and J. Schoenmakers. Projected particle methods for solving McKean- Vlasov stochastic differential equations.SIAM Journal on Numerical Analysis, 56(6):3169– 3195, 2018.doi:10.1137/17M1111024

    2018 doi

  35. [42]

    Notesonmeanfieldgames, 2013

    P.Cardaliaguet. Notesonmeanfieldgames, 2013. Pierre-LouisLions’lecturesatCollegede France. URL:https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf

    2013

  36. [43]

    Buckdahn, J

    R. Buckdahn, J. Li, S. Peng, and C. Rainer. Mean-field stochastic differential equations and associated PDEs, 2017.doi:10.1214/15-AOP1076

    2017 doi

  37. [44]

    Zhang.Backward Stochastic Differential Equations

    J. Zhang.Backward Stochastic Differential Equations. From Linear to Fully Nonlinear Theory. Number 1 in Probability Theory and Stochastic Modelling. Springer New York, 2017.doi:10.1007/978-1-4939-7256-2. 28

    2017 doi

  38. [45]

    dos Reis, G

    G. dos Reis, G. Smith, and P. Tankov. Importance sampling for McKean-Vlasov SDEs. Applied Mathematics and Computation, 453(31):128078, 2023.doi:10.1016/j.amc.2023. 128078

    2023 doi

  39. [46]

    Number 1 in Grundlehren der mathematischen Wissenschaften

    D.Bakry, I.Gentil, andM.Ledoux.Analysis and Geometry of Markov Diffusion Operators. Number 1 in Grundlehren der mathematischen Wissenschaften. Springer Cham, 2013. doi:10.1007/978-3-319-00227-9

    2013 doi

  40. [47]

    Dynamicalsystemsandcomplexnetworks: AKoopman operator perspective.Journal of Physics: Complexity, 5(4):041001, 2024.doi:10.1088/ 2632-072X/ad9e60

    S.KlusandN.DjurdjevacConrad. Dynamicalsystemsandcomplexnetworks: AKoopman operator perspective.Journal of Physics: Complexity, 5(4):041001, 2024.doi:10.1088/ 2632-072X/ad9e60

    2024

  41. [48]

    G. A. Pavliotis.Stochastic processes and applications. Diffusion Processes, the Fokker– Planck and Langevin Equations. Number 1 in Texts in Applied Mathematics. Springer New York, 2014.doi:10.1007/978-1-4939-1323-7

    2014 doi

  42. [50]

    Schütte, S

    C. Schütte, S. Klus, and C. Hartmann. Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning.Acta Numerica, 32:517–673, 2023.doi:10.1017/S0962492923000016

    2023 doi

  43. [51]

    Korda and I

    M. Korda and I. Mezić. On convergence of extended dynamic mode decomposition to the Koopman operator.Journal of Nonlinear Science, 28(2):687–710, 2018.doi:10.1007/ s00332-017-9423-0

    2018

  44. [52]

    G. W. Stewart and J. G. Sun.Matrix perturbation theory. Computer Science and Scientific Computing. Academic Press, 1990

    1990

  45. [53]

    Kato.Perturbation theory for linear operators

    T. Kato.Perturbation theory for linear operators. Number 2 in Classics in Mathematics. Springer Berlin, Heidelberg, Heidelberg, 1995.doi:10.1007/978-3-642-66282-9

    1995 doi

  46. [54]

    F. L. Bauer and C. T. Fike. Norms and exclusion theorems.Numerische Mathematik, 2:137–141, 1960.doi:10.1007/BF01386217

    1960 doi

  47. [55]

    Angeli, J

    L. Angeli, J. Barré, M. Kolodziejczyk, and M. Ottobre. Well-posedness and stationary solutions of McKean–Vlasov(S)PDEs.Journal of Mathematical Analysis and Applications, 526(2):127301, 2023.doi:10.1016/j.jmaa.2023.127301

    2023

  48. [56]

    M.Lipton, R.Mirollo, andS.H.Strogatz. TheKuramotomodelonasphere: Explainingits low-dimensional dynamics with group theory and hyperbolic geometry.Chaos: An Inter- disciplinary Journal of Nonlinear Science, 31(9):093113, 2021.doi:10.1063/5.0060233

    2021 doi

  49. [57]

    Marjanovic, M

    G. Marjanovic, M. J. Piggott, and V. Solo. Numerical methods for stochastic differential equations in the Stiefel manifold made simple. In2016 IEEE 55th Conference on Decision and Control (CDC), pages 2853–2860, 2016.doi:10.1109/CDC.2016.7798694

    2016

  50. [58]

    ReproducingkernelHilbertspaces in the mean field limit.Kinetic and Related Models, American Institute of Mathematical Sciences, 16(6):850–870, 2023.doi:10.3934/krm.2023010

    C.Fiedler, M.Herty, M.Rom, C.Segala, andS.Trimpe. ReproducingkernelHilbertspaces in the mean field limit.Kinetic and Related Models, American Institute of Mathematical Sciences, 16(6):850–870, 2023.doi:10.3934/krm.2023010. 29

    2023 doi

  51. [59]

    A. N. Shiryaev.Probability. Number 2 in Graduate Texts in Mathematics. Springer New York, 1996.doi:10.1007/978-1-4757-2539-1

    1996 doi

  52. [60]

    sup t∈[0,T]} W2(µt,ˆµt)2 # =E

    G. dos Reis, W. Salkeld, and J. Tugaut. Freidlin-Wentzell LDP in path space for McKean– Vlasov equations and the functional iterated logarithm law.The Annals of Applied Prob- ability, 29(3):1487–1540, 2019.doi:10.1214/18-AAP1416. A Auxiliary results Lemma A.1(Law of large numb...

    2019 doi

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.