Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations

Baoyou Qu; Chunrong Feng; Huaizhong Zhao

REVIEW 2 major objections 4 minor 27 references

Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Under a dissipativity condition, a stochastic differential equation with quasi-periodic coefficients has a unique random quasi-periodic path whose law is a quasi-periodic measure; lifting to a cylinder yields a unique ergodic invariant…

desk verdict New concepts and solid existence proofs, but the uniqueness proof for the invariant measure on the cylinder contains a false x-independent estimate and an unjustified Fatou step. read the letter →

arxiv 1908.10015 v2 pith:GNSG6WQA submitted 2019-08-27 math.PR math.DS

Chunrong Feng , Baoyou Qu , Huaizhong Zhao This is my paper · ORCID

classification math.PRmath.DS MSC 60H1037H1060J6037A30

keywords quasi-periodicmeasuresrandompathsstochasticdifferentialequationsinvariantergodicityFokker-PlanckequationdynamicalsystemsMarkoviansemigroups

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

This paper introduces random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups, then proves that for stochastic differential equations whose drift and diffusion are quasi-periodic in time there is a unique uniformly $L^p$-bounded random quasi-periodic path, provided a two-parameter periodic lift of the coefficients is uniformly dissipative and the noise is not too strong relative to the drift. The law of this path is the unique quasi-periodic entrance measure, and its density, when it exists, satisfies the Fokker-Planck equation. Lifting the flow to the cylinder $[0,\tau_1) \times [0,\tau_2) \times \mathbb{R}^d$ turns the quasi-periodic measure into a family of measures whose time average converges to a unique invariant measure, given by the double average of $\delta_{s_1} \times \delta_{s_2} \times \tilde{\rho}_{s_1,s_2}$, and this measure is ergodic. The paper matters because it provides a mathematical language for systems that combine randomness with two incommensurate periodicities, such as temperature or energy demand with daily and annual cycles.

What carries the argument

The central object is the two-variable periodic lift $(\tilde{b}, \tilde{\sigma})$ of the quasi-periodic coefficients, and the parameterised SDE $K^{r_1,r_2}$ it generates: for each pair $(r_1,r_2)$, $K^{r_1,r_2}(t,s,x)$ solves the SDE with coefficients $\tilde{b}(t+r_1,t+r_2,x)$ and $\tilde{\sigma}(t+r_1,t+r_2,x)$. The key identity (1.3), $K^{r_1,r_2}(t+r,s+r,x,\theta_{-r}\omega) = K^{r_1+r,r_2+r}(t,s,x,\omega)$, gives the family a shift-invariance that the original time-dependent SDE lacks. Under the dissipativity condition, each $K^{r_1,r_2}$ has a unique $L^p$ pull-back limit, and periodicity of the lift in $r_1$ and $r_2$ makes that limit a random quasi-periodic path. Lifting the flow to the cylinder then produces a cocycle whose time averages are tight; Birkhoff ergodic theory and the minimality of the two-torus rotation identify the limiting invariant measure.

What would settle it

Find a quasi-periodic drift and diffusion that satisfy the original one-sided dissipativity on the diagonal but for which every two-variable periodic lift violates the uniform dissipativity Condition 3.3, and check whether the SDE still has two distinct uniformly $L^p$-bounded random paths; if it does, the theorem's premise is not necessary, and if no such path exists at all, the premise is doing the work.

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Extended reading notes

Core claim

The central discovery is that quasi-periodicity of the coefficients can be converted into a family of autonomous-like SDEs indexed by two parameters, and the pull-back limit produces a unique random quasi-periodic path. More precisely, assuming the coefficients admit a two-variable periodic lift satisfying a uniform one-sided dissipativity condition with exponent $\alpha$ and Lipschitz noise with constant $\beta$, and $\alpha > (p-1)\beta^2/2$ for some $p \ge 2$, the SDE has a unique uniformly $L^p$-bounded random quasi-periodic path; its law is the unique quasi-periodic probability measure in $M_p$. When the lifted flow is considered on the cylinder $[0,\tau_1) \times [0,\tau_2) \times \mathbb{R}^d$, the averages of the lifted quasi-periodic measures are tight and converge to the unique invariant measure $\frac{1}{\tau_1\tau_2}\int_0^{\tau_1}\int_0^{\tau_2} \delta_{s_1} \times \delta_{s_2} \times \tilde{\rho}_{s_1,s_2}\, ds_1\, ds_2$, which is ergodic.

Load-bearing premise

The construction works only if the quasi-periodic coefficients admit a two-variable periodic lift that is uniformly dissipative (one-sided Lipschitz with a positive $\alpha$) and H\"older continuous, so that the parameterised family of SDEs is well behaved; quasi-periodicity of the original coefficients alone does not guarantee such a lift.

Editorial extensions

If this is right

Every SDE satisfying Conditions 3.2 and 3.3 with $\alpha > (p-1)\beta^2/2$ has a unique uniformly $L^p$-bounded random quasi-periodic path, so quasi-periodic forcing does not destroy uniqueness of the long-time random behaviour.
The law of this path is the unique quasi-periodic measure in $M_p$, giving a direct bridge from pathwise random quasi-periodicity to measure-valued quasi-periodicity.
The lifted semigroup on the cylinder has a unique invariant measure, so the long-time statistical behaviour is ergodic and can be written explicitly as an average of $\delta_{s_1} \times \delta_{s_2} \times \tilde{\rho}_{s_1,s_2}$.
When the diffusion coefficient is invertible and the coefficients are smooth, the quasi-periodic measure has a density that satisfies the Fokker-Planck equation, and entrance measures correspond one-to-one to solutions of the infinite-horizon Fokker-Planck problem.
The abstract Theorem 3.18 extends the invariant-measure conclusion to any Feller Markovian semigroup with a tight family of averaged quasi-periodic entrance measures.

Reading between the lines

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

The explicit quasi-periodic Ornstein-Uhlenbeck example suggests a concrete statistical test: fit a quasi-periodic mean-reverting model to temperature or electricity-price data and compare the empirical long-run distribution with the predicted Gaussian quasi-periodic measure; the paper does not run such tests.
The construction should extend to any finite number of incommensurate periods, as the authors state, so the two-period case is a template for multi-period random quasi-periodicity; a natural next step is to check whether the invariant measure remains ergodic with three or more periods.
The uniqueness of the random quasi-periodic path is obtained before any perfection argument, which suggests that the core existence result is insensitive to pathwise regularity; one could attempt to weaken Condition 3.4 by proving continuity of the limiting path with a weaker growth condition.
Since the invariant measure is ergodic on the cylinder, observable long-time averages of smooth functions of the lifted process converge to the double average; this gives a quantitative prediction that could be tested numerically in the quasi-periodic OU model.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

This paper is worth reading: it gives the first definitions of random quasi-periodic paths and quasi-periodic measures for SDEs, and proves existence and uniqueness under a dissipativity condition. The machinery — the two-shift reparameterized SDE, the cylinder lift, the explicit O-U example — is genuinely new and mostly executed with care. Theorems 3.5 and 3.9 appear solid: contraction estimates, Lp bounds, tightness and the Feller property all check out.

But the proof of Theorem 3.16, uniqueness of the invariant measure on the cylinder, has a real gap. The authors assert that ||K^{r1,r2}(t,s,x) - tilde_phi^{r1,r2}(t)||_2 is bounded by C exp(-(alpha-beta^2/2)(t-s)) with C independent of x. That is false. In the O-U case with zero drift and identity noise, the distance is e^{-a(t-s)}|x|. So the error term in (3.58) is actually C(1+|x|^2)/delta^2 times a decaying exponential, and passing through Fatou in (3.59) requires the unknown invariant measure to have finite second moment. The paper never proves that. I suspect the claim is true and the gap is fixable via a standard Lyapunov estimate, but as written the uniqueness proof is incomplete.

The other soft spot is Section 4: Lemma 4.4, which is load-bearing for the density and Fokker-Planck results, is not proved; the paper defers to a self-cited arXiv preprint. That is worth flagging to a referee, though not fatal if the preprint is correct.

Minor issues: some Ito estimates omit localization details, and Theorem 4.7 bundles uniqueness of the infinite-horizon Fokker-Planck problem into an iff, which is awkward. These are minor.

Overall: the paper makes a real conceptual contribution, and the existence/uniqueness results for quasi-periodic paths and measures are likely correct. The invariant measure uniqueness needs a repaired proof, and the density section is conditional on an external lemma. This should go to peer review — a serious referee should engage with it — but the authors need to address the gap before publication.

Referee Report

2 major / 4 minor

Summary. The paper introduces notions of random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups, and applies them to SDEs with quasi-periodic coefficients. Assuming a two-variable periodic lift of the coefficients satisfying uniform one-sided dissipativity, Lipschitz, boundedness and Hölder conditions, it proves existence and uniqueness of uniformly L^p-bounded random quasi-periodic paths and quasi-periodic entrance measures in M_p. It then lifts the flow to the cylinder [0,τ1)×[0,τ2)×R^d, proves tightness of the Cesàro averages and the Feller property, and obtains an invariant measure. The paper further claims uniqueness and ergodicity of this invariant measure, and gives sufficient conditions for the existence of densities and for the Fokker-Planck equation. A multidimensional Ornstein-Uhlenbeck example with explicit quasi-periodic path and measure is worked out.

Significance. If the main theorems are correct, this is a valuable first systematic framework for random quasi-periodicity, parallel to the existing theory of random periodic paths and periodic measures. The explicit dissipativity and regularity assumptions make the existence results checkable, and the OU example gives concrete formulas that help validate the definitions. The paper also contains useful technical components: L^p contraction estimates, Kolmogorov continuity arguments, tightness, and the Feller property. The claimed uniqueness and ergodicity of the lifted invariant measure is the capstone of the paper; since the proof of that claim is currently incomplete, the significance is conditional on a repair. The main theorems also depend on the strong premise, stated in Conditions 3.2 and 3.3, that the quasi-periodic coefficients admit a two-variable periodic lift satisfying uniform dissipativity; the paper does not discuss how to verify this beyond the linear example.

major comments (2)

[Theorem 3.16, proof, equations (3.58)-(3.59)] The estimate displayed before (3.58), asserting that ||K^{r1,r2}(t,s,x)-tilde_phi^{r1,r2}(t)||_2 is bounded by C exp(-(alpha-beta^2/2)(t-s)) with C=C(alpha,beta,tilde_M) independent of x, is false. In the Ornstein-Uhlenbeck example with S=0, sigma=I, A=aI and r1=r2=0, one has K^{0,0}(0,-T,x)=e^{-aT}x+int_{-T}^0 e^{av} dW_v while tilde_phi^{0,0}(0)=int_{-infty}^0 e^{av} dW_v, so the L^2 distance equals e^{-aT}|x|. Consequently the term subtracted in (3.58) must carry a factor depending on |x|, typically 1+|x|^2, and passing to the limit inside the integral in (3.59) via Fatou's lemma requires a second-moment bound on the arbitrary invariant measure nu. The paper does not establish int |x|^2 nu(dx)<infty before using Fatou's lemma. The uniqueness conclusion of Theorem 3.16 is therefore not supported as written. This gap appears fixable by adding a Lyapunov/dissipativity argument for the invariant measure and by tracking the x-dependence in the contraction estimate, but the missing argument is load-bearing.
[Theorem 3.16, proof, Birkhoff step] The proof invokes equation (3.33) of Lemma 3.7 to obtain L^2 continuity of tilde_phi(t,s) and hence weak continuity of tilde_rho(t,s). However Lemma 3.7 is stated under Conditions 2.1, 3.2, 3.3 and 3.4, with the additional threshold assumptions such as p gamma >= 2 kappa. Theorem 3.16 assumes only Conditions 3.2 and 3.3 together with alpha > beta^2/2, and does not assume Condition 3.4 or the large-moment condition. Thus the continuity needed for the unique ergodicity argument is not established under the hypotheses of Theorem 3.16. The authors should either prove the required L^2 continuity under Conditions 3.2-3.3 alone or add the missing assumptions to the statement of Theorem 3.16.

minor comments (4)

[Proof of Theorem 3.16, first paragraph] There is a typo in 'there exists a uniquen invariant probability measure'; it should read 'a unique invariant probability measure'.
[Section 3.3, after equation (3.45)] The identity 'tilde_mu_s = hat_mu_{s1,s2}' is not meaningful as written; it should be 'tilde_mu_s = hat_mu_{s,s}' or the variables s1 and s2 should be set equal to s in the definition of hat_mu.
[Lemma 4.4 and Theorem 4.5] The notation tilde P in Lemma 4.4 and in the proof of Theorem 4.5 is ambiguous: it appears to denote the measure with density equal to the stochastic exponential, but this is not defined explicitly. Please introduce a clear notation for the Girsanov-changed measure.
[Conditions 3.2 and 3.3] The paper would benefit from a discussion of how the uniform dissipativity of the two-variable lift can be verified for a given quasi-periodic pair (b,sigma). The OU example constructs the lift explicitly, but for general quasi-periodic coefficients the existence of a lift with a positive uniform alpha is a nontrivial premise that strongly restricts the class of equations covered.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; results are derived from explicit assumptions rather than assumed conclusions.

full rationale

The paper's derivation chain is self-contained. Under Conditions 3.2 and 3.3, the random quasi-periodic path is constructed as an L^p pullback limit of the parameterized SDE (3.6), and uniqueness follows from the dissipativity contraction estimate of Lemma 2.6. The quasi-periodic measure is then defined as the law of this path, with periodicity inherited from the periodic lift of the coefficients, not assumed. The invariant measure on the cylinder is obtained by proving tightness of the averages of the lifted quasi-periodic measures (Lemma 3.14) and identifying the weak limit via the Feller property and Birkhoff's ergodic theorem; uniqueness is attempted via a contraction estimate. There is no parameter fitting or renaming of a known result. The only self-citation that is used, Lemma 4.4 adapted from the authors' prior work [13], concerns a Girsanov-type absolute continuity lemma and is not load-bearing for the main quasi-periodicity theorems. (A separate correctness concern: the contraction estimate in Theorem 3.16 asserts a constant C independent of x, which fails for simple OU examples; this affects the proof of uniqueness of the invariant measure, but it is an error in the argument rather than a circularity.)

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces new mathematical objects (random quasi-periodic paths, quasi-periodic measures, and the cylinder lift) but these are definitions and constructions, not physically postulated entities. No free parameters are fitted to data; the constants alpha, beta, M, tau_1, tau_2 are model inputs. The most consequential axiom is the existence of a well-behaved two-variable periodic lift of the coefficients, which is not automatically implied by quasi-periodicity of the original coefficients. The proofs also rely on standard stochastic analysis tools.

assumptions (8)

standard math Standard Ito calculus, including Ito's formula for powers of the solution and L^p estimates for stochastic integrals.
Used throughout in Lemma 2.5, Lemma 2.6, Lemma 3.7 and the proofs of Theorems 2.4 and 3.5.
standard math Kolmogorov continuity criterion for pathwise continuity of the solution and random path.
Invoked in Lemma 3.7 to prove continuity of u, K, phi, and tilde_phi under the stated moment conditions.
standard math Girsanov theorem and BMO martingale theory for exponential martingales.
Used in Lemma 4.3 and Lemma 4.4 to prove absolute continuity of the laws of the original and noise-only SDEs.
standard math Birkhoff ergodic theorem and unique ergodicity of minimal rotations on the two-torus.
Used in Theorem 3.16 to identify the limit of the empirical averages of lifted measures and to prove uniqueness of the invariant measure.
domain assumption Condition 2.1: Uniform one-sided Lipschitz dissipativity of the drift and Lipschitz continuity of the diffusion coefficient, with uniform boundedness at zero.
This is the main dissipativity assumption guaranteeing existence and uniqueness of the random path and entrance measure.
domain assumption Condition 3.2 and 3.3: Existence of a two-variable periodic lift of the coefficients with uniform dissipativity, Lipschitz, and Holder properties in both time variables.
This is the load-bearing assumption for the reparameterized SDE (3.6) and the entire quasi-periodic construction. It is stronger than mere quasi-periodicity of the original coefficients.
domain assumption The reciprocals of the periods tau_1 and tau_2 are rationally linearly independent.
This incommensurability assumption is required for the minimal rotation and unique ergodicity on the cylinder in Theorem 3.16.
domain assumption Conditions 4.1 and 4.6: Uniform invertibility of the diffusion coefficient and global boundedness and Holder regularity of the coefficients and their derivatives.
These are assumed in Section 4 to obtain the density of the entrance measure and the Fokker-Planck equation.

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Pith. "Pith review of Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations." pith.science (2026). https://pith.science/paper/GNSG6WQA

@misc{pith2026190810015,
  author       = {Pith},
  title        = {Pith review of: Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GNSG6WQA}},
  note         = {Machine review of arXiv:1908.10015}
}

read the original abstract

In this paper, we define random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups. We give a sufficient condition for the existence and uniqueness of random quasi-periodic paths and quasi-periodic measures for stochastic differential equations and a sufficient condition for the density of the quasi-periodic measure to exist and to satisfy the Fokker-Planck equation. We obtain an invariant measure by considering lifted flow and semigroup on cylinder and the tightness of the average of lifted quasi-periodic measures. We further prove that the invariant measure is unique, and thus ergodic.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Arnold, Random Dynamical Systems , Springer-Verlag Berlin Heidelberg New York, 1998

L. Arnold, Random Dynamical Systems , Springer-Verlag Berlin Heidelberg New York, 1998

work page 1998

[2] [2]

V. I. Arnold, Proof of a Theorem by A. N. Kolmogorov on the i nvariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russian Math Survey , Vol. 18 (1963), 9-36

work page 1963

[3] [3]

P. W. Bates, K.N. Lu and B.X. Wang, Attractors of non-auto nomous stochastic lattice systems in weighted spaces, Physica D , Vol. 289 (2014), 32-50

work page 2014

[4] [4]

F. E. Benth and J. Saltyte-Benth, The volatility of tempe rature and pricing of weather derivatives, Quan- titative Finance , Vol. 7 (2007), 553-561

work page 2007

[5] [5]

F. E. Benth, J. Kallsen and T. Meyer-Brandis, A Non-Gauss ian Ornstein-Uhlenbeck process for electricity spot price modelling and derivatives pricing, Applied Mathematical Finance , Vol. 14 (2007), 153-169. 36

work page 2007

[6] [6]

Chekroun, E

M. Chekroun, E. Simonnet and M. Ghil, Stochastic climate dynamics: random attractors and time- dependent invariant measures, Physica D , Vol. 240 (2011), 1685-1700

work page 2011

[7] [7]

A. M. Cherubini, J. S W Lamb, M. Rasmussen and Y. Sato, A ran dom dynamical systems perspective on stochastic resonance, Nonlinearity, Vol. 30 (2017), 2835-2853

work page 2017

[8] [8]

Da Prato and J

G. Da Prato and J. Zabczyk, Ergodicity for inﬁnite dimensional systems , London Mathematical Society Lecture Note Series, 229, Cambridge University Press, 1996

work page 1996