Long-time behaviour of dynamical systems driven by bounded mixing noises

Peng Gao; Sergei Kuksin

REVIEW 3 major objections 4 minor 21 references

Long-time behaviour of dynamical systems driven by bounded mixing noises

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Dissipative systems driven by bounded mixing noise converge to a unique statistical state exponentially fast.

desk verdict Finite-dimensional mixing results are solid and worth reading; the infinite-dimensional theorem, which carries the primitive-equations application, is not fully supported as written and needs a careful referee. read the letter →

arxiv 2607.05981 v2 pith:CWIP7VZY submitted 2026-07-07 math.DS math.PR

Peng Gao , Sergei Kuksin This is my paper · ORCID

classification math.DSmath.PR MSC 37A2537H3035Q3035Q5637L4035R60

keywords randomlyforcedequationmixingMarkovliftingKantorovichfunctionalnon-Markovdynamicsdual-Lipschitzdistancetotalvariationprimitiveequations

topics Navier-Stokes Regularity

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 proves that a broad class of dissipative random dynamical systems — including ODEs and the 3D primitive equations of atmospheric dynamics — driven by bounded random noises that mix in the past become exponentially mixing: the laws of any two solutions with different initial data converge to each other (and to a unique invariant statistical state) at an exponential rate. The key novelty is that the noise need not be stationary or Markovian; it only needs to have conditional distributions given its past that are Lipschitz in total variation, recurrent to zero, and non-degenerate in a finite-dimensional projection. The proof lifts the system to a Markov process on the infinite-dimensional history space and uses a Doeblin coupling with a Kantorovich functional to show exponential contraction. If correct, this gives a rigorous basis for statistical prediction in a wide range of randomly forced dissipative systems.

What carries the argument

The machinery is the Markovian lifting of the non-Markov system (1.1) to the product space X = X×K, where K is the space of past noise histories, via the transition probabilities P_k(U;·) = S_*(U, Q_k(ξ;·)), with Q_k the regular conditional distribution of the noise given its past. Hypothesis (SF) — that Q_k(ξ;·) is Lipschitz in ξ in total variation with respect to the weighted-history metric d(ξ,ξ') = Σ α^{|k|}∥ξ_k − ξ'_k∥ — makes the lifted process sufficiently regular for a Doeblin coupling. The coupling operators R_k, R'_k are built by combining a local 'stabilisation' step (using the Moore–Penrose right inverse of D_ηS to reduce the distance between two trajectories) with a maximal coup

What would settle it

For a concrete system meeting (SF),(RZ),(LCR),(GD) — for example S(u,η)=a u + η on a bounded interval with an i.i.d. noise smoothed by a Lipschitz moving average — numerically compute the total-variation or dual-Lipschitz distance between the laws of two solutions started at widely separated initial states as a function of time. The theorem predicts an exponential decay with a rate γ independent of the initial states; observing algebraic or non-uniform decay would refute the claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that under Hypotheses (SF), (RZ), (LCR), (GD) in finite dimensions — and (SF), (RZ), (DLP), (ALC), (GD) in infinite dimensions — system (1.1) is exponentially mixing: for initial states v,v' in X, ∥D(u_k(v)) − D(u_k(v'))∥_var ≤ C e^{−γk} in finite dimensions, and the dual-Lipschitz analogue in infinite dimensions. Moreover, there is a unique-in-law two-sided process {û_k} extending the dynamics in the sense D(û_k) = S_*(D(û_{k−1}, η̂_k)) with η̂ distributed as η. If the noise is stationary, {û_k} is the unique stationary process and its one-time marginals give the unique stationary measure. The proof works by lifting (1.1) to a Markov process on

Load-bearing premise

The entire result rests on the assumption (SF) that the noise's conditional distribution given its past moves by no more than a Lipschitz multiple of the past's weighted total-variation distance; if this fails, the coupling argument breaks down, and it visibly fails for many natural continuous-path noises unless the noise is re-encoded as (ν_k, η^0_k).

Editorial extensions

If this is right

For any two initial states in the invariant set X, the total-variation distance between their laws decays exponentially in time (finite-dimensional case), so long-run statistics are uniquely determined and computable.
There exists a unique two-sided process û_k satisfying the system in law that attracts all trajectories; when the noise is stationary, this yields a unique stationary measure and exponential mixing of the system.
For randomly forced ODEs driven by bounded mixing noises with continuous paths, the laws of any two solutions converge in dual-Lipschitz distance at an exponential rate for all times t ≥ 0 (Corollary 6.9).
The 3D primitive equations of atmospheric dynamics, driven by bounded continuous mixing noise, are exponentially mixing in the dual-Lipschitz metric (Theorem 6.10).
Any family of transition probabilities on noise histories satisfying (SF) and (RZ) defines a unique random process that is feebly exponentially mixing — a Dobrushin-type reconstruction theorem (Corollary 3.7).

Reading between the lines

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

The theorem's scope hinges on the (SF) condition, which fails for natural H^1_loc noises and forces the modification η̂_k = (ν_k, η^0_k); a systematic characterisation of noises admitting such a modification would reveal how broad the theory really is.
In infinite dimensions the authors only prove dual-Lipschitz mixing and explicitly doubt total-variation mixing; a natural test is to seek a concrete parabolic SPDE where TV mixing actually fails, to confirm the weaker metric is necessary.
Since the proof uses only the transition probabilities {Q_l} and not the process itself, the results can be read as a property of non-homogeneous Markov operators on the history space; this suggests potential applications to deterministic driving signals with suitable mixing properties.
The paper does not address mixing at continuous times beyond integer times for general systems; extending the rates to all real times with explicit constants would be a useful next step for the ODE application.

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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

The finite-dimensional part of this paper is in good shape and worth engaging with seriously. The genuinely new material is there: non-stationary bounded mixing noises, a construction that handles H^1_loc sample paths by replacing the noise with a modified process η̂_k, and a weakening of the density hypothesis in the infinite-dimensional statement. The Markov-lifting argument and the Kantorovich-functional coupling are presented clearly, and the applications to ODEs with continuous bounded noise are plausible. The authors are honest that the finite-dimensional theorem overlaps with [KS25] under slightly stronger assumptions; the added value is in the non-stationary formulation and the modified noise.

The soft spot is exactly where the stress-test note points: Section 5. The text says outright that the presentation is sketchy and that missing details may be extracted from [KS25]. Lemma 5.6 is deferred to a proposition in [KS25], and the rest of the proof is asserted to repeat earlier steps without checking that uniformity of constants survives the relaxed hypothesis that ∪F_n need not be dense. That matters, because Theorem 5.2 is the foundation for the primitive-equations application in Theorem 6.10. This is not a fatal flaw in the underlying idea — the finite-dimensional proof is largely explicit, and the infinite-dimensional extension probably works with enough work — but as written the central infinite-dimensional claim is unverified. The reliance on results from [KS25] and [KS26] without restatement is also heavy; a referee should check whether those constants and estimates carry over unchanged.

A minor but real caveat: Hypothesis (SF) fails for the natural H^1-continuous noise, and the paper fixes this by moving to the modified process η̂_k. That is legitimate, but it means the theorems apply only when such a modification exists and satisfies the assumptions. For the primitive equations, the verification is plausible but not fully detailed; a referee should push on this point too.

Who gets value from this paper? People working on random dynamical systems and statistical hydrodynamics, especially those interested in non-Markovian bounded noise. The finite-dimensional result and the examples are a genuine contribution. But I would not cite the infinite-dimensional theorem in its current state without checking the deferred details, and I would not trust the primitive-equations claim until Section 5 is written out properly.

Recommendation: send it to peer review, absolutely. A serious referee can separate the solid finite-dimensional core from the sketchy infinite-dimensional part. The paper deserves referee time and probably a major revision, not a desk reject.

Referee Report

3 major / 4 minor

Summary. The paper studies dissipative discrete-time random dynamical systems u_k = S(u_{k-1}, η_k) driven by bounded random noises whose regular conditional distributions with respect to the past satisfy Lipschitz, recurrence and non-degeneracy conditions. It lifts the dynamics to a Markov process on the history space X = X × K and proves exponential mixing in total variation in finite dimensions (Theorem 3.3), exponential mixing in the dual-Lipschitz metric in infinite dimensions (Theorem 5.2), and existence/uniqueness of a two-sided weak solution. Applications are given to ODEs driven by random processes with H^1_loc trajectories and to the randomly forced primitive equations (Theorem 6.10). The paper explicitly extends the framework of Kuksin–Shirikyan [KS25] by allowing non-stationary noises and by not assuming that the union of the finite-dimensional subspaces F_n is dense in E.

Significance. If the results are complete, they constitute a valuable extension of [KS25] to non-stationary bounded mixing noises and to continuous-time noises with H^1_loc sample paths, and they provide an exponential-mixing statement for the randomly forced primitive equations, a system of current interest. The Kantorovich-functional Doeblin coupling argument is natural and the finite-dimensional proof is mostly explicit. However, the manuscript itself states in Section 5 that the infinite-dimensional proof is only a sketch, and several key estimates in both the finite- and infinite-dimensional parts are deferred to [KS25] and [KS26]. Thus the advertised infinite-dimensional theorem is not fully established within the present text.

major comments (3)

[Section 5, Theorem 5.2 and the 'Sketch of the proof'] Theorem 5.2 is a central advertised result and is the basis of the primitive-equations application (Theorem 6.10), but its proof is not contained in the manuscript. The text explicitly says 'Our presentation below is sketchy', the key Lemma 5.6 is deferred to '[KS25, Proposition 6.1]', and the final paragraph asserts that the rest repeats Steps 3,4 of Section 4.1 and Sections 4.2,4.3 without verifying that the constants N, q, p, l, κ remain uniform under the relaxed hypothesis (ALC), where ∪F_n is not assumed dense in E. This is load-bearing: an undetected gap would leave the infinite-dimensional exponential-mixing claim unsupported. Please supply a complete proof or state and prove the precise external result used, checking that its hypotheses cover the present relaxation.
[Section 3.3, derivation of (3.7)] The finite-dimensional total-variation conclusion (3.7) depends on two imported ingredients: the assertion that S_*(u,(ξ^⊥_F, Q_{kF})) has a Lipschitz density, and the estimate (3.20). Both are taken from '[KS26, Theorem 4.1 and Corollary 4.2]' without restating the results or their hypotheses. Since (3.7) is the main total-variation mixing statement, the reader cannot verify from the present text that all assumptions of [KS26] hold in this setting. Please restate these results, or give a self-contained proof.
[Lemma 5.5, proof, Step 1] The proof of Lemma 5.5 says 'Since A(y)(F_∞) is dense in H', but the lemma's assumption is only that the closure of A(y)(F_∞) contains the finite-dimensional subspace G. The intended Dini-theorem argument can be made to work on the compact set Y × B_G(1) using the weaker assumption, but as written the proof is internally inconsistent. This is a correctable issue, but it is part of the already sketchy infinite-dimensional proof and should be fixed explicitly.

minor comments (4)

[Abstract and Introduction] Typographical and grammatical issues: 'Under a linearised controllability assumptions', repeated '(η1)(η1)(η1)' labels, 'continuos', 'aslo', 'provied', and similar. These should be corrected.
[Section 6.5] Notation is confusing: 'H^1(R1_loc;H)' should presumably be H^1_loc(R;H), and the spaces H^1, H^1_0 used there are not defined consistently with Section 6.3. Please clarify.
[Section 6.3, Hypothesis (H)] Hypothesis (H) is essentially the whole content of the applicability of the main theorems to continuous-time noises. The paper gives constructions showing that such processes exist, but it would help to state explicitly that Hypothesis (H) is an assumption and not a consequence of the preceding construction.
[Corollary 3.7] The corollary is interesting and, if correct, gives a Dobrushin-style reconstruction of a process from its conditional laws. A brief comparison with the precise statement in [Dob70] would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

Minor self-citation/dependency burden: no constructional circularity, but Section 5's infinite-dimensional proof is explicitly deferred to [KS25] and Section 3.3 imports [KS26] for the TV upgrade.

full rationale

I walked the paper's claimed derivation chain and found no step in which a stated conclusion is used to define its own hypothesis or in which a fitted parameter is renamed as a prediction. The hypotheses (SF), (RZ), (LCR)/(DLP), (ALC), (GD) are conditions on the noise's regular conditional distributions, recurrence, densities, and on the deterministic map's controllability/dissipation; the exponential-mixing estimates are derived via the Markov lifting of Section 2 and the Doeblin-coupling/Kantorovich argument of Sections 4/5, not assumed. Corollary 3.7 is a genuine application of Theorem 3.5 to the trivial system S≡0, not a renaming. The applications in Section 6 are conditional on the stated Hypothesis (H), so no output is built into the input. The main caveats are proof dependencies: Section 5 states 'Our presentation below is sketchy, but missing details may be extracted from the work [KS25]', Lemma 5.6's proof is 'carried out by repeating that of Lemma 4.3', and Lemma 4.3 itself invokes '[KS25, Proposition 6.1]'; Section 3.3 relies on '[KS26, Theorem 4.1]' and 'Corollary 4.2 in [KS26]' for the total-variation upgrade. These are overlapping-author citations, and the paper explicitly relaxes [KS25]'s density assumption ('we do not assume that the space ∪F_n is dense in E'), so the infinite-dimensional theorem is not self-contained as written. This is a deductive-gap/dependency burden, not a circular equivalence, hence the low score.

Assumptions & free parameters 0 free parameters · 8 assumptions · 2 invented entities

The ledger shows the paper is an extension of an existing program: it postulates strong structural conditions on the noise (SF, RZ, LCR/DLP/ALC, H) and imports key technical propositions from prior Kuksin–Shirikyan papers instead of re-proving them. The only invented objects are auxiliary constructions in the proof.

assumptions (8)

domain assumption S:H×E→H is C^2, bounded on bounded sets, S(X×K)⊂X, and the unforced system satisfies (GD): ∥S^k(u;0)∥≤a∥u∥ with a<1.
Section 1 and Hypothesis (GD), Section 3.1. Defines the dissipative compact regime on which all mixing estimates are made.
domain assumption η_k takes values in a compact set K, X⊂H is compact and contains the origin, and the probability space is isomorphic to [0,1] with Lebesgue measure.
Section 1 and Notation and Agreements. Compactness and Polish structure are used throughout.
domain assumption (SF): Q_l(ξ;·) can be chosen Lipschitz in ξ in total variation on (K,d), uniformly in l.
Section 1, Hypothesis (SF). The paper later notes the natural H^1 continuous process violates this, forcing the modified noise η̂_k in Section 6.3.
domain assumption (RZ): uniformly in l,ξ, there is positive probability of entering the δ-neighborhood of 0 and staying there for n steps.
Section 1, Hypothesis (RZ). Used in Proposition 4.5 to produce a uniform hitting probability.
domain assumption (LCR)(a,b): existence of a finite-dimensional F⊂E with D_ηS(u,η):F→H surjective near X×K, and uniformly Lipschitz conditional densities ρ_lF.
Section 3.1. Basis of the local stabilisation Lemma 4.2 and the total-variation density argument in Section 3.3.
domain assumption Infinite-dimensional counterparts (DLP) and (ALC): increasing finite-dimensional F_n, a determining subspace G⊂H, and density of D_ηS(F_∞) over G.
Section 5. Replaces LCR when dim H=∞; the paper weakens [KS25] by not requiring ∪F_n dense in E.
standard math External results imported as black boxes: [KS25, Proposition 6.1], [KS26, Theorem 4.1, Corollary 4.2].
Used at Lemmas 4.3, 5.6 and Section 3.3. They are not restated in this preprint, and [KS25]/[KS26] share an author with this paper.
domain assumption For applications: Hypotheses (K), (LC), (H), and deterministic well-posedness/compactness results for the primitive equations from [B22].
Sections 6.4–6.5. Restrict the class of equations and noises to which Theorems 3.3 and 5.2 are applied.

invented entities (2)

History lifting U_k=(u_k, η^k) on X×K
purpose: Turn the non-Markovian system (1.1) into a Markov process on an infinite-dimensional history space so that Doeblin coupling can be applied.
Introduced in Section 2; purely a proof device with no observable content.
Modified noise η̂_k=(ν_k, η^0_k)
purpose: Lift H^1_loc continuous trajectories to satisfy Hypothesis (SF) by separating endpoint values ν_k from the interior H^1_0 component.
Section 6.3; a change of variables in the input process, not a new physical entity.

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Pith. "Pith review of Long-time behaviour of dynamical systems driven by bounded mixing noises." pith.science (2026). https://pith.science/paper/CWIP7VZY

@misc{pith2026260705981,
  author       = {Pith},
  title        = {Pith review of: Long-time behaviour of dynamical systems driven by bounded mixing noises},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWIP7VZY}},
  note         = {Machine review of arXiv:2607.05981}
}

read the original abstract

We study the mixing properties of discrete-time and continuous-time dissipative dynamical systems driven by bounded mixing random forces. The continuous-time systems are reduced to discrete-time random dynamical systems generated by time-one maps, so that the main analysis is carried out in the discrete setting. We introduce a class of mixing random forcings whose regular conditional distributions with respect to the past satisfy natural regularity, recurrence, and non-degeneracy assumptions, extending the framework previously developed for more restrictive classes of processes in a paper by Kuksin-Shirikyan in GAFA (2025). Under a linearised controllability assumptions on the system, we prove exponential mixing in the total variation metric for finite-dimensional phase spaces. We then establish an infinite-dimensional counterpart yielding exponential mixing in the dual-Lipschitz metric under suitable amendments of restrictions on the system and the random forcing. Our approach is based on lifting the dynamics to an appropriate Markov process on an infinite-dimensional history space and applying a Doeblin coupling argument through the method of Kantorovich functional. As applications, we derive exponential mixing for a broad class of ordinary differential equations driven by mixing random processes with bounded continuous trajectories. As an application of our result to PDEs we discuss the randomly perturbed primitive equations of atmospheric dynamics.

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

V. I. Bogachev and I. I. Malofeev, Kantorovich problems and conditional measures depending on a parameter, J. Math. Anal. Appl. 486 (2020), no. 1, 123883, 30

2020

[2] [2]

A. A. Borovkov, Ergodicity and Stability of Stochastic Processes , John Wiley & Sons, Ltd., Chichester, 1998

1998

[3] [3]

Boulvard, Mixing for the primitive equations under bounded non-degenerate noise, Stochastic and Partial Differential Equations: Analysis and Computations 10 (2022): 126-159

P.-M. Boulvard, Mixing for the primitive equations under bounded non-degenerate noise, Stochastic and Partial Differential Equations: Analysis and Computations 10 (2022): 126-159

2022

[4] [4]

Boulvard, P

P.-M. Boulvard, P. Gao and V. Nersesyan, Controllability and ergodicity of three dimensional primitive equations driven by a finite-dimensional force, Archive for Rational Mechanics and Analysis 247 (2023): no. 1, Article 2

2023

[5] [5]

Dedecker, P

J. Dedecker, P. Doukhan, G. Lang, J. R. Le \'o n, S. Louhichi, and C. Prieur, Weak D ependence: with E xamples and A pplications , Springer, New York, 2007

2007

[6] [6]

R. L. Dobru s in, Prescribing a system of random variables by conditional distributions, Teor. Verojatnost. i Primenen. 15 (1970), no. 3, 458--486

1970

[7] [7]

R. M. Dudley, Real A nalysis and P robability , Cambridge University Press, Cambridge, 2002

2002

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Eckmann, C.-A

J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet , Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Comm. Math. Phys. 201 (1999), 657--697

1999