arXiv:2608.00370v1

Continuity of measure-theoretic entropy for stochastic differential equations

Zhenxin Liu, Lixin Zhang

math.DSmath.PR28D2037A1037A3560H10

Paper metadata

Original submission: August 1, 2026
Claimed MSC: 28D20, 37A10, 37A35, 60H10
arXiv categories: math.DS, math.PR
Paper license: arXiv non-exclusive distribution license ↗
Comments: 19 pages, no figures

Abstract

For stochastic differential equations, we establish a relationship between the measure-theoretic entropy of the stochastic flow and the rate of volume growth of stable submanifolds under iteration. By combining the result of Kifer and Yomdin (1988), we show that under a suitable integrability condition, for systems with $C^{\infty}$ coefficients, the measure-theoretic entropy is upper semicontinuous with respect to the coefficients of SDEs.

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Audit summary

Audited against arXiv v1

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 findingsContains wrong statements

The main upper-semicontinuity theorem is false under the stated $C^0$ convergence hypothesis. A rescaled compactly supported deterministic flow gives coefficients converging uniformly to zero while preserving positive metric entropy and concentrating its invariant measure at a zero-entropy fixed point. The two application theorems impose additional recurrence and nondegeneracy hypotheses, so this counterexample does not disprove them, but their printed proofs depend on the invalid general theorem and do not independently establish their conclusions.

Theorem 3.7Incorrect

Uniform convergence of coefficients does not imply upper semicontinuity of entropy

Pages 14–15 · Theorem 3.7 · arXiv:2608.00370v1

Choose a smooth compactly supported vector field $X$ on some $\mathbb R^d$ whose flow has a compact invariant probability measure $\nu$ with positive time-one entropy; for example, embed a smooth suspension of a horseshoe in Euclidean space and extend its vector field through a tubular neighborhood with compact support. For $\varepsilon_k\downarrow0$ , set $A_k(x)=\varepsilon_kX(x/\varepsilon_k)$ and $B_k=0$ , while $A=B=0$ . The flow satisfies $\Phi_k^t(x)=\varepsilon_k\Phi_X^t(x/\varepsilon_k)$ . Hence $\lVert A_k-A\rVert_{C^0}\to0$ , every coefficient is $C^\infty$ with all derivatives bounded for each fixed $k$ , and every integral in (3.26) is finite. If $\mu_k$ is the pushforward of $\nu$ by $x\mapsto\varepsilon_kx$ , then $\mu_k\Rightarrow\delta_0$ and $h_{\mu_k}(\Phi_k)=h_\nu(\Phi_X)>0$ , whereas the limiting zero flow has $h_{\delta_0}(\Phi)=0$ . All printed hypotheses hold, but the asserted inequality fails. A valid theorem needs substantially stronger, uniform smooth control than $C^0$ convergence alone.

Full paper, version 1 ↗

Theorems 4.2 and 4.4Not able to verify

The application statements are not independently established

Pages 16–18 · Theorems 4.2 and 4.4 · arXiv:2608.00370v1

These theorems add a uniform Lyapunov condition and nondegeneracy of the diffusion on the relevant set, so the deterministic rescaling counterexample to Theorem 3.7 does not directly apply. Their proofs, however, consist of invoking Theorem 3.7 after Proposition 4.1 or Lemma 4.3. Since Theorem 3.7 is false and no replacement argument exploiting the additional hypotheses is supplied, a nontrivial proof obligation remains: one must derive entropy control uniform in $k$ , rather than merely finiteness of every fixed system's smooth norms. No proof or counterexample for these narrower statements is supplied here.

02Proofs5 reported findingsContains incorrect or incomplete proofs

The proof of the main theorem cannot be repaired under its printed hypotheses because the theorem has an explicit counterexample. Two earlier arguments also contain independent substantive gaps: Lemma 3.3 uses a false entropy identity for a fixed partition, and Theorem 3.5 treats merely measurable Oseledets data as continuous on an arbitrary compact set and omits the zero-positive-exponent case. Two mechanical notation errors are reported separately in yellow.

Proof of Theorem 3.7Incorrect as written

The argument lacks the uniform smooth control excluded by the counterexample

Page 15 · Equations (3.27)–(3.29) · arXiv:2608.00370v1

Equation (3.29) makes the local-volume term vanish separately for each $\Phi_k$ by allowing the differentiability order $r$ to tend to infinity, but the hypotheses contain no bound uniform in $k$ on the higher derivatives or on the associated local complexity. The rescaled-flow construction in the statements finding satisfies (3.26) for every fixed $k$ while retaining positive entropy, so the asserted conclusion cannot follow. Repair classification: No repair under the stated hypotheses; the theorem must be strengthened by a uniform topology or explicit uniform tail-entropy estimate.

Lemma 3.3Incomplete as written

The proof uses a false fixed-partition entropy identity

Pages 6–8 · display following Equation (3.6) · arXiv:2608.00370v1

The proof writes $h_\mu(\Phi,\alpha)=N^{-1}h_\mu(\Phi^N,\alpha)$ for the same fixed partition $\alpha$ . In general only the system entropy scales as $h_\mu(\Phi^N)=Nh_\mu(\Phi)$ ; for a fixed partition the correct $N$ -block relation uses $\bigvee_{j=0}^{N-1}\Phi^{-j}\alpha$ . Substituting that block changes the conditional-partition argument and must be propagated through (3.1)–(3.6). Repair classification: Plausible repair only, by giving the standard local-entropy proof with the full $N$ -block partition; the manuscript does not do so.

Theorem 3.5Incomplete as written

The Lyapunov-chart construction omits measurable-selection and exponent cases

Pages 11–14 · proof of Theorem 3.5, especially Equations (3.20)–(3.21) · arXiv:2608.00370v1

The proof first infers from $r(\Phi,\epsilon,\Lambda)>0$ that a positive Lyapunov exponent exists, which does not follow from Ruelle's inequality. It then asserts that the Oseledets bundles and the measurable chart-size and angle functions are continuous on the arbitrary compact set $\Lambda$ for almost every noise realization. Oseledets theory supplies measurability, while a Lusin argument gives continuity only after restricting to a suitably chosen large-measure compact subset and controlling varying multiplicities. Consequently the positive infima defining $M_2(\omega)$ and the subsequent finite covering are not established in the claimed scope. Repair classification: No repair supplied; the missing exponent case and a correct Pesin-block selection must be incorporated throughout the volume comparison.

Definition of $r(\Phi,n,\delta,\epsilon,\Lambda,x,\omega)$Typo

The separated-pair formula repeats the base point

Page 6 · paragraph before Lemma 3.3 · arXiv:2608.00370v1

For distinct $y,z\in F_n(x,\omega)$ , the printed separation condition compares $\Phi^i(x,\omega)$ with $\Phi^i(y,\omega)$ . Replace $x,y$ in that comparison by $y,z$ . The surrounding phrase 'for any $y\ne z$ ' and every later use as an $(n,\delta)$ -separated set determine this correction uniquely.

Definition of $\ln^+$Typo

The positive logarithm is printed as a positive part of its argument

Page 11 · definition immediately following $\overline G_{\mathrm{loc}}$ · arXiv:2608.00370v1

The paper prints $\ln^+x=\max\{1,x\}$ . Replace this by $\ln^+x=\max\{0,\ln x\}$ . The notation, the logarithmic growth-rate formulas, and the integrability conditions determine the intended correction mechanically; leaving the printed definition literal would invalidate the stated growth normalization.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.00370v1
Authors listed: Zhenxin Liu, Lixin Zhang
Audit date: August 18, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. 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. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

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P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

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Continuity of measure-theoretic entropy for stochastic differential equations

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Uniform convergence of coefficients does not imply upper semicontinuity of entropy

The application statements are not independently established

The argument lacks the uniform smooth control excluded by the counterexample

The proof uses a false fixed-partition entropy identity

The Lyapunov-chart construction omits measurable-selection and exponent cases

The separated-pair formula repeats the base point

The positive logarithm is printed as a positive part of its argument

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