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arxiv: 2604.25717 · v1 · submitted 2026-04-28 · 🧮 math.NA · cs.NA· math.PR

Splitting AVF method for generalized Langevin equations: probability density function and geometric ergodicity

Xinjie Dai , Xingyu Liu , Diancong Jin , Liying Sun This is my paper

Pith reviewed 2026-05-07 15:09 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.PR
keywords generalized Langevin equationssplitting methodsaveraged vector fieldprobability density functiongeometric ergodicityMalliavin calculusnumerical SDEsquasi-Markovian systems
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The pith

The splitting AVF method for generalized Langevin equations yields a smooth probability density function converging at first order and establishes geometric ergodicity of the numerical solution.

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

The paper proposes a structure-preserving splitting averaged vector field method for quasi-Markovian generalized Langevin equations to address challenges from superquadratic potentials and degenerate noise. It shows that this method keeps the exponential integrability, Malliavin differentiability, and ergodicity of the original system. Using these preserved properties, the authors prove that the numerical solution has an existing and smooth probability density function that approximates the exact one with first-order accuracy. They further demonstrate geometric ergodicity by checking Lyapunov and minorization conditions with a localized approach. This is useful for reliable long-time simulations of systems with memory effects in nonequilibrium dynamics.

Core claim

By constructing a splitting averaged vector field method that preserves exponential integrability, Malliavin differentiability, and uniform non-degeneracy, the numerical solution to the quasi-Markovian generalized Langevin equation is shown to possess a smooth probability density function. This density converges to that of the exact solution with first-order accuracy. The geometric ergodicity of the numerical solution is established by verifying the Lyapunov condition and the minorization condition through a localized technique.

What carries the argument

Splitting averaged vector field (AVF) discretization of the quasi-Markovian GLE that simultaneously retains exponential integrability, Malliavin differentiability, and uniform non-degeneracy.

If this is right

  • The probability density function of the numerical solution exists and is smooth.
  • It converges to the exact solution's probability density with first-order accuracy.
  • The numerical solution satisfies the geometric ergodicity property.
  • Numerical experiments validate the theoretical preservation of these properties.

Where Pith is reading between the lines

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

  • The method may allow for better long-term statistical sampling in molecular simulations involving memory kernels.
  • Similar splitting strategies could be applied to other degenerate stochastic systems to prove density regularity.
  • The localized technique for ergodicity might simplify proofs in related numerical methods for SDEs.

Load-bearing premise

The chosen splitting and AVF scheme must maintain the quasi-Markovian structure along with uniform non-degeneracy of the numerical solution for superquadratic potentials.

What would settle it

A specific superquadratic potential and splitting choice where the Malliavin covariance matrix of the numerical solution becomes degenerate, preventing the existence of a smooth PDF.

Figures

Figures reproduced from arXiv: 2604.25717 by Diancong Jin, Liying Sun, Xingyu Liu, Xinjie Dai.

Figure 1
Figure 1. Figure 1: The theoretical framework of main results. For the numerical study of the GLE (1.2) with superquadratic potentials and degenerate noise, the main contributions of this paper are summarized below: (i) We present a novel splitting AVF method for the GLE (1.2), which is proven to achieve first￾order strong convergence, building upon its exponential integrability. (ii) Based on Malliavin differentiability and … view at source ↗
Figure 2
Figure 2. Figure 2: 3D surface plots of the distribution at different times t = 2, 16, 128, 512 view at source ↗
Figure 3
Figure 3. Figure 3: 2D contour plots of the distribution at different times t = 2, 16, 128, 512 view at source ↗
Figure 4
Figure 4. Figure 4: Temporal averages 1 N PN n=1 E[g(Yn)] for four different initial values view at source ↗
read the original abstract

The generalized Langevin equation (GLE) constitutes a fundamental model for describing nonequilibrium dynamics with memory effects. To overcome the numerical challenges arising from superquadratically growing potentials and degenerate noise, we propose and analyze a structure-preserving splitting averaged vector field (AVF) method for a quasi-Markovian GLE. The core advantage of this method lies in its ability to simultaneously preserve the exponential integrability, Malliavin differentiability, and ergodicity of the underlying continuous system. Notably, by leveraging exponential integrability, Malliavin differentiability, and uniform non-degeneracy of the numerical solution, we obtain the existence and smoothness of its probability density function, which converges to that of the exact solution with first-order accuracy. Furthermore, by validating the Lyapunov condition and the minorization condition using a localized technique, we establish the geometric ergodicity of the numerical solution. Finally, numerical experiments are conducted to confirm the theoretical results.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The paper proposes a splitting averaged vector field (AVF) method for quasi-Markovian generalized Langevin equations (GLEs) with superquadratic potentials. It claims the scheme preserves exponential integrability, Malliavin differentiability, and ergodicity of the continuous system. Using these, the authors establish existence and smoothness of the numerical probability density function (PDF) with first-order convergence to the exact PDF. Geometric ergodicity of the numerical solution is proved via localized Lyapunov and minorization conditions, with numerical experiments confirming the results.

Significance. If the preservation of uniform non-degeneracy and the resulting PDF convergence and geometric ergodicity hold, the work would contribute rigorous analysis for structure-preserving integrators applied to hypoelliptic SDEs with memory. This is relevant for long-time accurate simulations in statistical mechanics and molecular dynamics. The integration of Malliavin calculus with splitting AVF for superquadratic cases represents a technical advance, provided the key estimates are robust.

major comments (2)
  1. Abstract: The central claims of PDF existence, smoothness, and first-order convergence rest on uniform non-degeneracy of the Malliavin covariance matrix for the split AVF solution. For superquadratic V(x), the splitting of the memory kernel and force terms may fail to yield a lower bound uniform in time and |x|, causing the hypoelliptic regularity argument to collapse. Explicit estimates or a proof that non-degeneracy holds after splitting (beyond the quasi-Markovian structure) are required, as this is load-bearing for all PDF-related results.
  2. Geometric ergodicity section (likely §5): The localized Lyapunov/minorization technique inherits the same uniform non-degeneracy requirement. If the Malliavin covariance can degenerate for large |x|, the minorization condition may not hold uniformly, undermining the geometric ergodicity claim. More detailed localization bounds or verification for the specific splitting would be needed to make this independent of the PDF analysis.
minor comments (1)
  1. Numerical experiments: The abstract mentions experiments confirming the theory, but without reported error bars, quantitative convergence rates, or details on how superquadratic potentials are tested, it is difficult to assess empirical support for the first-order PDF convergence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below.

read point-by-point responses

  1. Referee: Abstract: The central claims of PDF existence, smoothness, and first-order convergence rest on uniform non-degeneracy of the Malliavin covariance matrix for the split AVF solution. For superquadratic V(x), the splitting of the memory kernel and force terms may fail to yield a lower bound uniform in time and |x|, causing the hypoelliptic regularity argument to collapse. Explicit estimates or a proof that non-degeneracy holds after splitting (beyond the quasi-Markovian structure) are required, as this is load-bearing for all PDF-related results.

    Authors: We thank the referee for identifying this key technical point. The splitting AVF scheme is constructed to inherit the Malliavin differentiability and exponential integrability from the continuous quasi-Markovian GLE. These properties are used in Section 3 to derive a lower bound on the smallest eigenvalue of the Malliavin covariance matrix that is uniform in time and in the state variable, even when the potential is superquadratic; the bound follows from moment estimates controlled by the preserved exponential integrability and the specific form of the AVF discretization on the force and memory terms. The quasi-Markovian structure is essential but is combined with the splitting to obtain the uniformity. To make the argument fully explicit and address the concern, we will insert additional intermediate estimates and a dedicated lemma on the non-degeneracy constant in the revised manuscript. revision: yes

  2. Referee: Geometric ergodicity section (likely §5): The localized Lyapunov/minorization technique inherits the same uniform non-degeneracy requirement. If the Malliavin covariance can degenerate for large |x|, the minorization condition may not hold uniformly, undermining the geometric ergodicity claim. More detailed localization bounds or verification for the specific splitting would be needed to make this independent of the PDF analysis.

    Authors: We appreciate the referee's observation on the dependence between the two parts of the analysis. In Section 5 the geometric ergodicity proof proceeds via a localized Lyapunov function whose growth is controlled by the exponential integrability preserved by the scheme; the minorization condition is then verified on compact sets whose size is chosen independently of the non-degeneracy constant. Because the non-degeneracy lower bound established in Section 3 is already uniform, the localization does not require further adjustment for large |x|. Nevertheless, to strengthen the presentation and make the independence from the PDF analysis clearer, we will add a short paragraph that explicitly verifies the minorization constant for the splitting AVF operator on the localized sets. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent stochastic analysis

full rationale

The paper establishes exponential integrability, Malliavin differentiability, and uniform non-degeneracy for its splitting AVF scheme, then applies standard Malliavin calculus for PDF existence/smoothness/convergence and localized Lyapunov/minorization for geometric ergodicity. These steps do not reduce by the paper's own equations to fitted parameters or self-referential definitions; the claims are grounded in external techniques applied to the proposed method without load-bearing self-citation chains or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; no explicit free parameters or invented entities are introduced. Axioms are standard background assumptions for SDE analysis.

axioms (2)
  • domain assumption The continuous quasi-Markovian GLE admits solutions possessing exponential integrability and Malliavin differentiability
    Invoked to transfer these properties to the numerical scheme.
  • ad hoc to paper Uniform non-degeneracy of the numerical solution holds after splitting
    Used to guarantee existence and smoothness of the probability density function.

pith-pipeline@v0.9.0 · 5471 in / 1487 out tokens · 67885 ms · 2026-05-07T15:09:41.639665+00:00 · methodology

discussion (0)

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Works this paper leans on

37 extracted references · 5 canonical work pages

  1. [1]

    Abdulle, G

    A. Abdulle, G. Vilmart, and K. C. Zygalakis , Long time accuracy of Lie-Trotter splitting methods for Lan gevin dynamics, SIAM J. Numer. Anal., 53 (2015), no. 1, 1–16

    2015

  2. [2]

    J. Bao, J. Shao, and C. Yuan , Approximation of invariant measures for regime-switching diffusions , Potential Anal., 44 (2016), pp. 707–727

    2016

  3. [3]

    Bena¨ım, S

    M. Bena¨ım, S. Le Borgne, F. Malrieu, and P.-A. Zitt , Qualitative properties of certain piecewise deterministi c Markov processes, Ann. Inst. Henri Poincar’e Probab. Stat., 51 (2015), no. 3, 1040–1075

    2015

  4. [4]

    C. Chen, T. Dang, J. Hong, and F. Zhang , A new class of splitting methods that preserve ergodicity an d exponential integrability for the stochastic Langevin equ ation, SIAM J. Numer. Anal., 63 (2025), pp. 1000–1024

    2025

  5. [5]

    J. Cui, J. Hong, and D. Sheng , Density function of numerical solution of splitting A VF sch eme for stochastic Langevin equation , Math. Comp., 91 (2022), pp. 2283–2333

    2022

  6. [6]

    J. Cui, J. Hong, and D. Sheng , Approximations to Probabilistic Characteristics of Stoch astic Differential Equa- tions, Springer, Singapore, (2026)

    2026

  7. [7]

    L. Dai, Y. Jiang, and X. W ang , An explicit splitting SA V scheme for the kinetic Langevin dy namics, arXiv preprint, (2025), arXiv: 2509.04251

    work page arXiv 2025

  8. [8]

    Dai and X

    L. Dai and X. W ang , Perturbation estimates for order-one strong approximatio ns of SDEs without globally mono- tone coefficients , IMA J. Numer. Anal., (2025), DOI: 10.1093/imanum/draf034

    work page doi:10.1093/imanum/draf034 2025

  9. [9]

    Dai and D

    X. Dai and D. Jin , Density functions for the overdamped generalized Langevin equation and its Euler–Maruyama method: smoothness and convergence , arXiv preprint, (2024), arXiv: 2405.11744

    work page arXiv 2024

  10. [10]

    M. H. Duong and H. D. Nguyen , Asymptotic analysis for the generalized Langevin equation with singular poten- tials, J. Nonlinear Sci., 34 (2024), Paper No. 62

    2024

  11. [11]

    M. H. Duong and X. Shang , Accurate and robust splitting methods for the generalized L angevin equation with a positive Prony series memory kernel , J. Comput. Phys., 464 (2022), Paper No. 111332

    2022

  12. [12]

    N. E. Glatt-Holtz, D. P. Herzog, S. A. McKinley, and H. D. Nguy en, The generalized Langevin equation with power-law memory in a nonlinear potential well , Nonlinearity, 33 (2020), pp. 2820–2852

    2020

  13. [13]

    D. P. Herzog and J. C. Mattingly , Ergodicity and Lyapunov functions for Langevin dynamics wi th singular potentials, Comm. Pure Appl. Math., 72 (2019), pp. 2231–2255

    2019

  14. [14]

    J. Hong, D. Jin, and D. Sheng , Density convergence of a fully discrete finite difference meth od for stochastic Cahn–Hilliard equation , Math. Comp., 93 (2024), pp. 2215–2264

    2024

  15. [15]

    Hong and L

    J. Hong and L. Sun , Symplectic Integration of Stochastic Hamiltonian Systems , Springer, Singapore, (2022)

    2022

  16. [16]

    J. Hong, L. Sun, and X. W ang , High order conformal symplectic and ergodic schemes for the stochastic Langevin equation via generating functions , SIAM J. Numer. Anal., 55 (2017), pp. 3006–3029

    2017

  17. [17]

    Hong and X

    J. Hong and X. W ang , Invariant Measures for Stochastic Nonlinear Schr¨ odinger Equations: Numerical Approxi- mations and Symplectic Structures , Springer, Singapore, (2019)

    2019

  18. [18]

    Huang and X

    S. Huang and X. Li , Discretization, uniform-in-time estimations and approxi mation of invariant measures for nonlinear stochastic differential equations with non-unifo rm dissipativity, arXiv preprint, (2025), arXiv: 2511.12124

    work page arXiv 2025

  19. [19]

    Hutzenthaler, A

    M. Hutzenthaler, A. Jentzen, and X. W ang , Exponential integrability properties of numerical approx imation processes for nonlinear stochastic differential equations , Math. Comp., 87 (2018), pp. 1353–1413

    2018

  20. [20]

    Leimkuhler, C

    B. Leimkuhler, C. Matthews, and G. Stoltz , The computation of averages from equilibrium and nonequili brium Langevin molecular dynamics , IMA J. Numer. Anal., 36 (2016), 13–79

    2016

  21. [21]

    Leimkuhler and M

    B. Leimkuhler and M. Sachs , Ergodic properties of quasi-Markovian generalized Langev in equations with config- uration dependent noise and non-conservative force , Stochastic Dynamics out of Equilibrium, Springer Proceed ings in Mathematics & Statistics, Springer, Cham, 282 (2019), pp . 282–330

    2019

  22. [22]

    Leimkuhler and M

    B. Leimkuhler and M. Sachs , Efficient numerical algorithms for the generalized Langevin equation, SIAM J. Sci. Comput., 44 (2022), pp. A364–A388

    2022

  23. [23]

    L. Li, M. W ang, and Y. W ang , Estimates of the numerical density for stochastic differenti al equations with multiplicative noise , Sci. China Math., (2026), DOI: 10.1007/s11425-025-2496- 0

    work page doi:10.1007/s11425-025-2496- 2026

  24. [24]

    X. Li, X. Mao, and G. Yin , Explicit numerical approximations for stochastic different ial equations in finite and infinite horizons: truncation methods, convergence in pth moment and stability , IMA J. Numer. Anal., 39 (2019), pp. 847–892

    2019

  25. [25]

    S. H. Lim, J. Wehr, and M. Lewenstein , Homogenization for generalized Langevin equations with ap plications to anomalous diffusion , Ann. Henri Poincar´ e, 21 (2020), pp. 1813–1871

    2020

  26. [26]

    W. Liu, X. Mao, and Y. Wu , The backward Euler–Maruyama method for invariant measures of stochastic differ- ential equations with super-linear coefficients , Appl. Numer. Math., 184 (2023), pp. 137–150

    2023

  27. [27]

    Liu and Z

    Z. Liu and Z. Liu , Numerical unique ergodicity of monotone SDEs driven by nond egenerate multiplicative noise , J. Sci. Comput., 103 (2025), Paper No. 87. SPLITTING A VF METHOD FOR GLES 35

    2025

  28. [28]

    Lu and J

    Y. Lu and J. C. Mattingly , Geometric ergodicity of Langevin dynamics with Coulomb int eractions, Nonlinearity, 33 (2020), pp. 675–699

    2020

  29. [29]

    Mao and L

    X. Mao and L. Szpruch , Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative- type stochastic differential equations with super-linear di ffusion coefficients , Stochastics, 85 (2013), pp. 144–171

    2013

  30. [30]

    J. C. Mattingly, A. M. Stuart, and D. J. Higham , Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise , Stochastic Process. Appl., 101 (2002), pp. 185–232

    2002

  31. [31]

    G. N. Milstein and M. V. Tretyakov , Computing ergodic limits for Langevin equations , Phys. D, 229 (2007), 81–95

    2007

  32. [32]

    Nualart , The Malliavin Calculus and Related Topics , Probability and its Applications (New York), Springer- Verlag, Berlin, second ed., (2006)

    D. Nualart , The Malliavin Calculus and Related Topics , Probability and its Applications (New York), Springer- Verlag, Berlin, second ed., (2006)

    2006

  33. [33]

    P ang, X

    C. P ang, X. W ang, and Y. Wu , Projected Langevin Monte Carlo algorithms in non-convex an d super-linear setting, J. Comput. Phys., 526 (2025), Paper No. 113754

    2025

  34. [34]

    G. A. P avliotis , Stochastic Processes and Applications: Diffusion Processes , the Fokker–Planck and Langevin Equations, Springer, New York, 2014

    2014

  35. [35]

    G. R. W. Quispel and D. I. McLaren , A new class of energy-preserving numerical integration met hods, J. Phys. A, 41 (2008), Paper No. 045206

    2008

  36. [36]

    Talay, Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme , Markov Process

    D. Talay, Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme , Markov Process. Related Fields, 8 (2002), pp. 163–198

    2002

  37. [37]

    X. Wu, B. R. Brooks, and E. V anden-Eijnden , Self-guided Langevin dynamics via generalized Langevin eq uation, J. Comput. Chem., 37 (2016), 595–601. School of Mathematics and Statistics, Yunnan University, K unming 650500, Yunnan, China Email address : dxj@ynu.edu.cn School of Mathematics and Statistics, Yunnan University, K unming 650500, Yunnan, China ...

    2016