arXiv:2606.15843v2

Long-time Behaviour of DLRA for SDEs

Jianhai Bao, Haitao Wang, Yue Wu

math.PRmath.NA60H1058J6565C3037A2560J60

Paper metadata

Original submission: June 14, 2026
Latest revision: June 18, 2026 · version 2
Claimed MSC: 60H10, 58J65, 65C30, 37A25, 60J60
arXiv categories: math.PR, math.NA
Paper license: arXiv non-exclusive distribution license ↗

Abstract

We study dynamical orthogonal (DO) approximations of stochastic differential equations and investigate their long-time behaviour. The DO formulation represents the solution by a low-rank decomposition and leads to a coupled system consisting of an evolution equation on the Stiefel manifold and a reduced stochastic process. We establish the well-posedness of the strong DO system and derive quantitative error estimates between the original stochastic differential equation and its low-rank approximation in the Wasserstein distance. Our main contribution is the analysis of invariant probability measures for the DO dynamics. Under suitable dissipativity, Lipschitz continuity, and non-degeneracy assumptions on the coefficients, we prove the existence of an invariant probability measure for the strong DO system. The proof combines uniform moment estimates, a Krylov--Bogoliubov argument for an associated frozen system, and a Kakutani-Fan-Glicksberg fixed-point theorem to recover the self-consistent dynamics. We further show that the induced low-rank process admits an invariant probability measure and discuss the structure of invariant measures through several illustrative examples. These results provide a rigorous foundation for the use of dynamical low-rank approximations in the approximation of long-time statistical properties of stochastic dynamical systems.

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements2 reported findingsCorrect

The well-posedness, moment, Wasserstein-stability, and invariant-measure conclusions are correct under the stated hypotheses. The invariant-measure proof incorrectly treats a Wasserstein-type construction on finite signed measures as a normed locally convex space, but its conclusion has a direct verified repair using the weak topology on finite signed measures.

Theorems 3.2 and 4.5Correct

The DLRA dynamics and Wasserstein estimates follow from the stated Lipschitz and dissipativity assumptions

Pages 12–28 · Sections 3 and 4 · arXiv:2606.15843v2

The projected coefficients retain the required growth and local Lipschitz control, the stopping and moment estimates prevent explosion, and the synchronous coupling closes the displayed Gronwall inequalities. The Wasserstein bounds then follow by taking the infimum over initial couplings. The constants and moment orders used in the invariant-set construction are compatible with the later compactness argument.

Theorem 5.5Correct

An invariant low-rank law exists despite the invalid signed-measure ambient topology in the printed proof

Pages 29–32 · invariant-measure theorem · arXiv:2606.15843v2

The moment estimate makes the displayed set of probability laws tight, convex, and closed under weak convergence. Embed it in the locally convex vector space of finite signed measures with the weak topology. Prokhorov compactness gives a compact convex set, and the uniform moment bound upgrades weak convergence there to the $W_2$ convergence needed by the paper's continuity estimate. The Markov operator is affine and has closed graph, so the Kakutani–Fan–Glicksberg argument applies in this corrected ambient space and proves the same conclusion without changing the theorem's assumptions.

02Proofs2 reported findingsContains incorrect or incomplete proofs

Most proofs are correct. The proof of Theorem 5.5 incorrectly claims that its Wasserstein-type construction makes the space of finite signed measures a normed locally convex vector space before invoking a fixed-point theorem. The theorem is nevertheless repairable by using the weak topology on finite signed measures. One notation mismatch is a harmless typo.

Proof of Theorem 5.5Incorrect as written · verified repair

The displayed Wasserstein construction on signed measures is not a vector-space norm

Pages 29–32 · fixed-point argument · arXiv:2606.15843v2

The proof first enlarges the probability measures to a space $\mathcal P_2$ of finite signed measures and then states that its displayed $W_2$ construction makes $(\mathcal P_2,W_2)$ a normed locally convex Hausdorff topological vector space. This is false. Common constant shifts in the admissible dual functions make the displayed supremum infinite when the two signed measures have different total masses; even on equal positive masses, the square-root transport expression has square-root rather than degree-one homogeneity under positive scalar multiplication. It is therefore not a vector-space norm, so the cited Kakutani–Fan–Glicksberg theorem cannot be applied in that ambient topology as written. The verified repair is to place the compact convex family inside the vector space of finite signed measures with the weak topology. On this family, the uniform moment bound of order $p>2$ makes weak convergence imply $W_2$ convergence, so the compactness and closed-graph argument proved in the paper remain valid and the same fixed-point theorem applies.

Last paragraph of Theorem 5.5Typo

The semigroup parameter uses the wrong measure symbol once

Page 31 · sentence beginning `We will show' · arXiv:2606.15843v2

The prose says that the limit law is invariant for $P_t(\nu)$ , while the displayed calculation and the theorem consistently use $P_t(\mu)$ . Replacing $\nu$ by $\mu$ is uniquely determined by the surrounding formulas and has no effect on the result.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.15843v2
Authors listed: Jianhai Bao, Haitao Wang, Yue Wu
Audit date: August 18, 2026

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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

is proved for every sufficiently small

delta>0

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P

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P(delta)

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delta>0

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t>0

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Long-time Behaviour of DLRA for SDEs

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The DLRA dynamics and Wasserstein estimates follow from the stated Lipschitz and dissipativity assumptions

An invariant low-rank law exists despite the invalid signed-measure ambient topology in the printed proof

The displayed Wasserstein construction on signed measures is not a vector-space norm

The semigroup parameter uses the wrong measure symbol once

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