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arxiv: 1907.02825 · v1 · pith:Y3GQKHN3new · submitted 2019-07-05 · 🧮 math.NA · cs.NA

Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems based on the Wong--Zakai approximation

Chuchu Chen , Jialin Hong , Chuying Huang This is my paper

Pith reviewed 2026-05-25 02:03 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords stochastic modified equationsrough Hamiltonian systemsWong-Zakai approximationsymplectic methodsrough path theorypathwise convergencebackward error analysis
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The pith

For symplectic methods on rough Hamiltonian systems, the associated stochastic modified equation based on the Wong-Zakai approximation has a Hamiltonian formulation.

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

The paper constructs a new stochastic modified equation as a perturbation of the Wong-Zakai approximation to the rough differential equation. For symplectic methods applied to rough Hamiltonian systems, this modified equation is shown to admit a Hamiltonian formulation. Pathwise convergence orders of the truncated modified equation to the numerical solution are established using techniques from rough path theory. When increments of the driving noise are replaced by truncated random variables, the one-step error of the modified equation is proved to be exponentially small in the time step size.

Core claim

For a symplectic method applied to a rough Hamiltonian system, the stochastic modified equation constructed as a perturbation of the Wong-Zakai approximation admits a Hamiltonian formulation. The pathwise convergence order of the truncated modified equation to the numerical method is obtained via rough path theory, and the one-step error becomes exponentially small when noise increments are simulated by truncated random variables.

What carries the argument

The stochastic modified equation defined via Wong-Zakai perturbation of the rough differential equation, which carries the Hamiltonian structure for symplectic methods.

Load-bearing premise

The Wong-Zakai approximation serves as a perturbation base that allows the stochastic modified equation to retain a Hamiltonian formulation.

What would settle it

A concrete counterexample in which a symplectic integrator on a rough Hamiltonian system produces a stochastic modified equation lacking Hamiltonian structure under the Wong-Zakai construction would falsify the claim.

Figures

Figures reproduced from arXiv: 1907.02825 by Chuchu Chen, Chuying Huang, Jialin Hong.

Figure 1
Figure 1. Figure 1: Mean-square error vs. Step size for Example 5.1 [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mean-square error vs. Step size for Example 5.2 [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of domains in the phase plane 32 [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The midpoint scheme (34) 0 10 20 30 40 50 0 10 20 Original 0 10 20 30 40 50 0 10 20 Error 2-modified 0 10 20 30 40 50 T 0 0.2 0.4 4-modified (a) Error for one trajectory 0 10 20 30 40 50 0 20 40 ExpRK 0 10 20 30 40 50 0 0.5 1 Energy error 10-13 2-modified 0 10 20 30 40 50 T 0 20 40 4-modified (b) Energy error for one trajectory [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The explicit RK method (35) 33 [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The symplectic partitioned RK method (36) [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗
read the original abstract

We investigate the stochastic modified equation which plays an important role in the stochastic backward error analysis for explaining the mathematical mechanism of a numerical method. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation, which is a perturbation of the Wong--Zakai approximation of the rough differential equation. For a symplectic method applied to a rough Hamiltonian system, the associated stochastic modified equation is proved to have a Hamiltonian formulation. Second, the pathwise convergence order of the truncated modified equation to the numerical method is obtained by techniques in the rough path theory. Third, if increments of noises are simulated by truncated random variables, we show that the one-step error can be made exponentially small with respect to the time step size. Numerical experiments verify our 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

0 major / 3 minor

Summary. The paper constructs a new stochastic modified equation (SME) for symplectic numerical methods on rough Hamiltonian systems by perturbing the Wong-Zakai approximation of the underlying rough differential equation. It proves that this SME admits a Hamiltonian formulation. Pathwise convergence orders of the truncated SME to the numerical solution are derived via rough path theory. When noise increments are replaced by truncated random variables, the one-step error is shown to be exponentially small in the step size. Numerical experiments are included to support the theoretical claims.

Significance. If the central proofs hold, the work supplies a backward-error-analysis tool for symplectic integrators under rough driving signals, which may clarify structure preservation over long times. The explicit Hamiltonian property of the modified equation and the pathwise convergence results obtained from rough-path techniques constitute concrete technical contributions. Reproducible numerical verification is also provided.

minor comments (3)
  1. [§2.3] §2.3: the precise definition of the perturbation term added to the Wong-Zakai equation should be displayed as a numbered display equation rather than inline, to facilitate later references in the convergence proof.
  2. [Figure 3] Figure 3: the legend does not distinguish the three curves by line style or marker; this reduces readability of the error plots.
  3. [Theorem 4.2] The statement of Theorem 4.2 should explicitly list the required regularity index of the rough path lift, rather than referring only to 'standard assumptions'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claims involve constructing a stochastic modified equation as a perturbation of the Wong-Zakai approximation for rough Hamiltonian systems, proving a Hamiltonian formulation for symplectic methods, and deriving pathwise convergence orders via rough path theory. These steps are presented as independent applications of existing rough path techniques without any quoted reductions to fitted inputs, self-definitional loops, or load-bearing self-citations that collapse the result to the paper's own equations by construction. The derivation chain remains self-contained against external mathematical frameworks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; no free parameters, invented entities, or ad-hoc axioms are mentioned in the summary text.

axioms (1)
  • domain assumption Rough path theory supplies the necessary lift and convergence tools for the Wong-Zakai approximation and pathwise error bounds.
    Invoked to obtain the stated convergence orders and Hamiltonian structure.

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24 extracted references · 24 canonical work pages · 1 internal anchor

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