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arxiv: 2503.13932 · v4 · pith:Z52QCYAUnew · submitted 2025-03-18 · 🧮 math.DS · math.CA

Most Probable KAM Tori in Stochastic Hamiltonian Systems

Xinze Zhang , Yong Li This is my paper

Pith reviewed 2026-05-22 23:59 UTC · model grok-4.3

classification 🧮 math.DS math.CA
keywords KAM theorystochastic Hamiltonian systemsmost probable pathOnsager-Machlup functionallarge deviation principlequasi-periodic toristochastic perturbations
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The pith

Stochastic noise preserves KAM tori as the most probable paths in Hamiltonian systems.

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

The paper establishes that quasi-periodic invariant tori from classical KAM theory persist in stochastic Hamiltonian systems when viewed as the most probable trajectories under noise. This matters because it shows that the underlying integrable structure selects specific paths that dominate the dynamics even when randomness is present. The work derives the Onsager-Machlup functional for systems with time-dependent noise coefficients and proves that this functional equals the large deviation rate function, which quantifies how trajectories deviate. A reader would care because the result supplies both a stability statement for the tori and an explicit way to measure fluctuations around them.

Core claim

The paper proves that under stochastic noise the original quasi-periodic invariant tori persist in the sense of the most probable path, thereby demonstrating the stability of KAM structures in random environments. It derives the Onsager-Machlup functional for stochastic Hamiltonian systems driven by time-dependent noise coefficients and shows that this functional coincides exactly with the large deviation rate function, providing a quantitative characterization of both the structural persistence of quasi-periodic motions and the geometry of fluctuations.

What carries the argument

The most probable path, identified by minimizing the Onsager-Machlup functional that equals the large deviation rate function for system trajectories.

If this is right

  • Quasi-periodic motions remain the trajectories of highest probability under the given stochastic perturbations.
  • Deviations from the tori, including rare events, are governed by the explicit large deviation rate function.
  • The stability result applies to Hamiltonian systems whose noise coefficients depend explicitly on time.
  • The coincidence of the Onsager-Machlup functional and the rate function supplies a direct link between most probable behavior and fluctuation geometry.

Where Pith is reading between the lines

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

  • Simulations of specific stochastic maps or flows could verify the result by sampling paths and confirming that the highest-probability ones stay near the classical tori.
  • The same identification of functionals may extend the approach to nearly integrable stochastic systems outside the Hamiltonian class.
  • The quantitative rate function could be used to estimate the probability of noise-induced escapes from neighborhoods of the tori.

Load-bearing premise

The Onsager-Machlup functional can be derived for stochastic Hamiltonian systems with time-dependent noise and a large deviation principle holds with an explicit rate function.

What would settle it

Numerical computation of the trajectory that minimizes the Onsager-Machlup functional in a concrete stochastic Hamiltonian system, such as a perturbed stochastic pendulum, showing systematic deviation from the unperturbed KAM torus.

Figures

Figures reproduced from arXiv: 2503.13932 by Xinze Zhang, Yong Li.

Figure 1
Figure 1. Figure 1: Comparison of phase-space trajectories among the deterministic (black bold solid line), stochastically perturbed [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The first row of three plots represents the phase space trajectories of the solutions to the stochastic nearly integrable [PITH_FULL_IMAGE:figures/full_fig_p036_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The comparison figure of Fig [PITH_FULL_IMAGE:figures/full_fig_p037_3.png] view at source ↗
read the original abstract

This paper investigates in depth how stochastic perturbations affect the integrable structure of Hamiltonian systems and develops a KAM theory for stochastic Hamiltonian dynamics, in the sense of the most probable path. We first derive the Onsager-Machlup functional for stochastic Hamiltonian systems driven by time-dependent noise coefficients and identify the most probable path of the system trajectories. Building on this, we establish a large deviation principle and obtain an explicit rate function that quantitatively characterizes trajectory deviations, in particular for rare events. The main contribution of this work is to prove that, under stochastic noise, the original quasi-periodic invariant tori persist in the sense of the most probable path, thereby demonstrating the stability of KAM structures in random environments. Moreover, we show that the Onsager-Machlup functional coincides exactly with the large deviation rate function, thereby providing a quantitative characterization of both the structural persistence of quasi-periodic motions and the geometry of fluctuations in stochastic Hamiltonian systems. Overall, our results extend the classical KAM framework to stochastic settings and offer new insight into the behavior of complex dynamical systems under noise.

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

1 major / 1 minor

Summary. The paper derives the Onsager-Machlup functional for stochastic Hamiltonian systems driven by time-dependent noise coefficients, identifies the most probable paths, establishes a large deviation principle with an explicit rate function, and proves that the original quasi-periodic KAM tori persist as minimizers of this functional (i.e., as most probable paths). It further claims that the Onsager-Machlup functional coincides exactly with the large-deviation rate function.

Significance. If the central derivation and identification hold with the required regularity, the work would provide a quantitative extension of KAM theory to stochastic settings by linking persistence of invariant tori to the geometry of large deviations. The explicit coincidence between the Onsager-Machlup functional and the rate function would be a notable strength, offering a variational characterization of both structural stability and fluctuation geometry in random Hamiltonian systems.

major comments (1)
  1. [Abstract / derivation of Onsager-Machlup functional] The derivation of the Onsager-Machlup functional for time-dependent noise coefficients (central to the abstract and the persistence claim) requires explicit verification of regularity conditions such as uniform bounds on the diffusion matrix and its derivatives, or a Stratonovich-to-Itô correction that preserves Hamiltonian structure. Standard derivations assume time-independent or uniformly elliptic diffusion; without these controls established in the manuscript, the identification of the functional with the large-deviation rate function lacks foundation and the persistence result has no rigorous basis.
minor comments (1)
  1. The abstract states the main results at a high level; the manuscript would benefit from a dedicated section or theorem statement that isolates the precise regularity hypotheses used for the time-dependent case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract / derivation of Onsager-Machlup functional] The derivation of the Onsager-Machlup functional for time-dependent noise coefficients (central to the abstract and the persistence claim) requires explicit verification of regularity conditions such as uniform bounds on the diffusion matrix and its derivatives, or a Stratonovich-to-Itô correction that preserves Hamiltonian structure. Standard derivations assume time-independent or uniformly elliptic diffusion; without these controls established in the manuscript, the identification of the functional with the large-deviation rate function lacks foundation and the persistence result has no rigorous basis.

    Authors: We thank the referee for this observation. Our derivation in Section 2 begins from the Stratonovich formulation of the stochastic Hamiltonian system and converts to Itô form, with the correction term explicitly computed as a time-dependent gradient that preserves the Hamiltonian structure. Assumption 2.1 states that the time-dependent diffusion coefficients are bounded, C^2-smooth, and uniformly elliptic. These conditions suffice for the Girsanov change of measure and the subsequent large-deviation analysis. Nevertheless, we acknowledge that the bounds on the derivatives of the diffusion matrix could be stated more explicitly. In the revised manuscript we will insert a new lemma (Lemma 2.3) that derives these uniform bounds directly from Assumption 2.1 and verifies that they remain valid for the time-dependent case, thereby placing the identification of the Onsager-Machlup functional with the rate function on a fully rigorous footing. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the Onsager-Machlup functional for time-dependent noise in stochastic Hamiltonian systems, establishes a large deviation principle with explicit rate function, and proves that the functional coincides with the rate function while showing persistence of KAM tori along most probable paths. These steps are presented as sequential mathematical derivations and identifications rather than reductions by definition or fitted inputs renamed as predictions. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation are indicated in the abstract or reader's summary. The central claims rest on explicit derivations and a large deviation principle whose assumptions are stated separately, making the chain independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract indicates reliance on deriving the Onsager-Machlup functional and establishing a large deviation principle for the specific class of systems, but provides no explicit free parameters, invented entities, or detailed axioms beyond standard domain assumptions in stochastic dynamics.

axioms (2)
  • domain assumption Stochastic Hamiltonian systems driven by time-dependent noise coefficients admit an Onsager-Machlup functional
    Stated as the first step in the abstract.
  • domain assumption A large deviation principle holds for the trajectories with an explicit rate function
    Established as part of the main contribution in the abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Most Probable KAM Tori in Stochastic Hamiltonian Systems Driven by Multiplicative Noise

    math.DS 2026-05 unverdicted novelty 7.0

    Proves persistence of most probable KAM tori under multiplicative noise in stochastic Hamiltonian systems and obtains the large-deviation rate function for trajectory deviations.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · cited by 1 Pith paper

  1. [1]

    Proof of a theorem of A

    Vladimir Igorevich Arnol’d. Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the hamiltonian. Russian Mathematical Surveys, 18(5):9, oct 1963

    work page 1963

  2. [2]

    Balakrishnan

    Alampallam V . Balakrishnan. Applied functional analysis, volume No. 3 ofApplications of Mathematics. Springer-Verlag, New York-Heidelberg, 1976

    work page 1976

  3. [3]

    Onsager-Machlup functional for stochastic evolution equations

    Xavier Bardina, Carles Rovira, and Samy Tindel. Onsager-Machlup functional for stochastic evolution equations. Ann. Inst. H. Poincaré Probab. Statist., 39(1):69–93, 2003

    work page 2003

  4. [4]

    Onsager–Machlup Functional for SLEκ Loop Measures

    Marco Carfagnini and Yilin Wang. Onsager–Machlup Functional for SLEκ Loop Measures. Comm. Math. Phys., 405(11):Paper No. 258, 2024

    work page 2024

  5. [5]

    The Onsager-Machlup function as Lagrangian for the most probable path of a jump-diffusion process.Nonlinearity, 32(10):3715–3741, 2019

    Ying Chao and Jinqiao Duan. The Onsager-Machlup function as Lagrangian for the most probable path of a jump-diffusion process.Nonlinearity, 32(10):3715–3741, 2019

    work page 2019

  6. [6]

    Persistence of invariant tori on submanifolds in Hamiltonian systems.J

    Shui-Nee Chow, Yong Li, and Yingfei Yi. Persistence of invariant tori on submanifolds in Hamiltonian systems.J. Nonlinear Sci., 12(6):585–617, 2002

    work page 2002

  7. [7]

    Constantine and Thomas H

    Gregory M. Constantine and Thomas H. Savits. A multivariate Faà di Bruno formula with applications. Trans. Amer. Math. Soc., 348(2):503–520, 1996

    work page 1996

  8. [8]

    Weak energy shaping for stochastic controlled port-Hamiltonian systems

    Francesco Cordoni, Luca Di Persio, and Riccardo Muradore. Weak energy shaping for stochastic controlled port-Hamiltonian systems. SIAM J. Control Optim., 61(5):2902–2926, 2023

    work page 2023

  9. [9]

    Newton’s method and periodic solutions of nonlinear wave equations

    Walter Craig and Clarence Eugene Wayne. Newton’s method and periodic solutions of nonlinear wave equations. Comm. Pure Appl. Math. , 46(11):1409–1498, 1993

    work page 1993

  10. [10]

    Sur un nouveau théorème-limite de la théorie des probabilités

    Harald Cramér. Sur un nouveau théorème-limite de la théorie des probabilités. Actualités Scientifiques et Industrielles, 736:5–23, 1938

    work page 1938

  11. [11]

    Rate of estimation for the stationary distribution of stochastic damping Hamiltonian systems with continuous observations

    Sylvain Delattre, Arnaud Gloter, and Nakahiro Yoshida. Rate of estimation for the stationary distribution of stochastic damping Hamiltonian systems with continuous observations. Ann. Inst. Henri Poincaré Probab. Stat., 58(4):1998–2028, 2022

    work page 1998

  12. [12]

    Large deviations techniques and applications, volume 38 of Applications of Mathematics (New York)

    Amir Dembo and Ofer Zeitouni. Large deviations techniques and applications, volume 38 of Applications of Mathematics (New York). Springer- Verlag, New York, second edition, 1998

    work page 1998

  13. [13]

    Jean-Dominique Deuschel and Daniel W. Stroock. Large deviations, volume 137 ofPure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1989

    work page 1989

  14. [14]

    M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. I. II. Comm. Pure Appl. Math., 28:1–47; ibid. 28 (1975), 279–301, 1975

    work page 1975

  15. [15]

    Donsker and Srinivasa R

    Monroe D. Donsker and Srinivasa R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. III. Comm. Pure Appl. Math., 29(4):389–461, 1976

    work page 1976

  16. [16]

    Donsker and Srinivasa R

    Monroe D. Donsker and Srinivasa R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. IV. Comm. Pure Appl. Math., 36(2):183–212, 1983

    work page 1983

  17. [17]

    Håkan Eliasson

    L. Håkan Eliasson. Perturbations of stable invariant tori for Hamiltonian systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 15(1):115–147, 1988

    work page 1988

  18. [18]

    Richard S. Ellis. Entropy, large deviations, and statistical mechanics, volume 271 of Grundlehren der mathematischen Wissenschaften [Funda- mental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1985

    work page 1985

  19. [19]

    Freidlin and Alexander D

    Mark I. Freidlin and Alexander D. Wentzell. Random perturbations of dynamical systems , volume 260 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, second edition, 1998. Translated from the 1979 Russian original by Joseph Szücs

    work page 1998

  20. [20]

    Measurable functions on Hilbert space

    Leonard Gross. Measurable functions on Hilbert space. Trans. Amer. Math. Soc., 105:372–390, 1962

    work page 1962

  21. [21]

    Limites approximatives sur l’espace de Wiener

    Gilles Hargé. Limites approximatives sur l’espace de Wiener. Potential Anal., 16(2):169–191, 2002

    work page 2002

  22. [22]

    Horn and Charles R

    Roger A. Horn and Charles R. Johnson. Matrix analysis. Cambridge University Press, Cambridge, 1985

    work page 1985

  23. [23]

    Stochastic differential equations and diffusion processes , volume 24 of North-Holland Mathematical Library

    Nobuyuki Ikeda and Shinzo Watanabe. Stochastic differential equations and diffusion processes , volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, second edition, 1989

    work page 1989

  24. [24]

    Ioannis Karatzas and Steven E. Shreve. Brownian motion and stochastic calculus, volume 113 ofGraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991

    work page 1991

  25. [25]

    On conservation of conditionally periodic motions for a small change in Hamilton’s function.Dokl

    Andre ˘i Nikolaevich Kolmogorov. On conservation of conditionally periodic motions for a small change in Hamilton’s function.Dokl. Akad. Nauk SSSR (N.S.), 98:527–530, 1954

    work page 1954

  26. [26]

    Koudjinan

    Comlan E. Koudjinan. A KAM theorem for finitely differentiable Hamiltonian systems. J. Differential Equations, 269(6):4720–4750, 2020

    work page 2020

  27. [27]

    Stochastic Hamiltonian dynamical systems

    Joan-Andreu Lázaro-Camí and Juan-Pablo Ortega. Stochastic Hamiltonian dynamical systems. Rep. Math. Phys., 61(1):65–122, 2008

    work page 2008

  28. [28]

    Gamma-limit of the Onsager-Machlup functional on the space of curves

    Tiejun Li and Xiaoguang Li. Gamma-limit of the Onsager-Machlup functional on the space of curves. SIAM J. Math. Anal., 53(1):1–31, 2021

    work page 2021

  29. [29]

    W. V . Li and Q.-M. Shao. Gaussian processes: inequalities, small ball probabilities and applications. InStochastic processes: theory and methods, volume 19 of Handbook of Statist., pages 533–597. North-Holland, Amsterdam, 2001

    work page 2001

  30. [30]

    An averaging principle for a completely integrable stochastic Hamiltonian system

    Xue-Mei Li. An averaging principle for a completely integrable stochastic Hamiltonian system. Nonlinearity, 21(4):803–822, 2008

    work page 2008

  31. [31]

    Persistence of lower dimensional tori of general types in Hamiltonian systems

    Yong Li and Yingfei Yi. Persistence of lower dimensional tori of general types in Hamiltonian systems. Trans. Amer. Math. Soc., 357(4):1565– 1600, 2005

    work page 2005

  32. [32]

    Fluctuations and irreversible process

    Stefan Machlup and Lars Onsager. Fluctuations and irreversible process. II. Systems with kinetic energy. Phys. Rev. (2), 91:1512–1515, 1953

    work page 1953

  33. [33]

    Onsager-Machlup functional for the fractional Brownian motion

    Sílvia Moret and David Nualart. Onsager-Machlup functional for the fractional Brownian motion. Probab. Theory Related Fields, 124(2):227– 260, 2002

    work page 2002

  34. [34]

    Jürgen K. Möser. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen, II, pages 1–20, 1962

    work page 1962

  35. [35]

    Fluctuations and irreversible processes

    Lars Onsager and Stefan Machlup. Fluctuations and irreversible processes. Phys. Rev. (2), 91:1505–1512, 1953

    work page 1953

  36. [36]

    Georgi St. Popov. KAM theorem for Gevrey Hamiltonians. Ergodic Theory Dynam. Systems, 24(5):1753–1786, 2004

    work page 2004

  37. [37]

    Integrability of Hamiltonian systems on Cantor sets

    Jürgen Pöschel. Integrability of Hamiltonian systems on Cantor sets. Comm. Pure Appl. Math., 35(5):653–696, 1982

    work page 1982

  38. [38]

    A KAM-theorem for some nonlinear partial differential equations

    Jürgen Pöschel. A KAM-theorem for some nonlinear partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , 23(1):119–148, 1996

    work page 1996

  39. [39]

    The Kolmogorov-Arnold-Moser theorem

    Dietmar Arno Salamon. The Kolmogorov-Arnold-Moser theorem. Math. Phys. Electron. J., 10:Paper 3, 37, 2004

    work page 2004

  40. [40]

    On the probability of large deviations of random variables

    Ivan N Sanov. On the probability of large deviations of random variables. United States Air Force, Office of Scientific Research, 1958. Most Probable KAM Tori in Stochastic Hamiltonian Systems 39

    work page 1958

  41. [41]

    Shepp and Ofer Zeitouni

    Larry A. Shepp and Ofer Zeitouni. A note on conditional exponential moments and Onsager-Machlup functionals. Ann. Probab., 20(2):652–654, 1992

    work page 1992

  42. [42]

    Stratonovi ˇc

    Ruslan L. Stratonovi ˇc. On the probability functional of diffusion processes. In Proc. Sixth All-Union Conf. Theory Prob. and Math. Statist. (Vilnius, 1960) (Russian), pages 471–482. Gosudarstv. Izdat. Politiˇcesk. i Nauˇcn. Lit. Litovsk. SSR, Vilnius, 1960

    work page 1960

  43. [43]

    Daniel W. Stroock. An introduction to the theory of large deviations. Universitext. Springer-Verlag, New York, 1984

    work page 1984

  44. [44]

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

    Denis Talay. Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Process. Related Fields, 8(2):163–198, 2002. Inhomogeneous random systems (Cergy-Pontoise, 2001)

    work page 2002

  45. [45]

    Fluctuations and irreversible thermodynamics

    Laszlo Tisza and Irwin Manning. Fluctuations and irreversible thermodynamics. Phys. Rev. (2), 105:1695–1705, 1957

    work page 1957

  46. [46]

    Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems

    Liming Wu. Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stochastic Process. Appl., 91(2):205–238, 2001

    work page 2001

  47. [47]

    Stochastic flows and Bismut formulas for stochastic Hamiltonian systems

    Xicheng Zhang. Stochastic flows and Bismut formulas for stochastic Hamiltonian systems. Stochastic Process. Appl., 120(10):1929–1949, 2010

    work page 1929

  48. [48]

    Onsager-machlup functional for stochastic differential equations with time-varying noise, 2024

    Xinze Zhang and Yong Li. Onsager-machlup functional for stochastic differential equations with time-varying noise, 2024

    work page 2024