arXiv:2208.08911v4

Uniform convergence of conditional distributions for one-dimensional diffusion processes

Guoman He, Hanjun Zhang

math.PR60J6060B1037A25

Paper metadata

Original submission: August 18, 2022
Latest revision: August 12, 2026 · version 4
Claimed MSC: 60J60, 60B10, 37A25
arXiv categories: math.PR
Paper license: arXiv non-exclusive distribution license ↗
Comments: 23 pages. Comments welcome!

Abstract

In this paper, we study the quasi-stationary behavior of the one-dimensional diffusion process with a regular or exit boundary at 0 and an entrance boundary at $\infty$ . By using the Doob's $h$ -transform, we show that the conditional distribution of the process converges to its unique quasi-stationary distribution exponentially fast in the total variation norm, uniformly with respect to the initial distribution. Moreover, we also use the same method to show that the conditional distribution of the process converges exponentially fast in the $ψ$ -norm to the unique quasi-stationary distribution. The rate of convergence of the conditional empirical measure to the quasi-ergodic distribution is also considered. Finally, two examples arising in population dynamics are also given to illustrate the main results.

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Audit summary

Audited against arXiv v4

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

01Statements3 reported findingsContains wrong statements

The weighted convergence estimate in Theorem 1.2 is not a valid statement for every initial distribution: its Radon-Nikodym derivative need not exist. The uniform total-variation theorem and the quasi-ergodic estimate are not able to be verified from the supplied arguments because two independent steps needed for uniformity over arbitrary initial laws are invalid.

Theorem 1.2Incorrect

The displayed bound is undefined for allowed initial distributions

Page 5 · Theorem 1.2 and the paragraph immediately preceding it · arXiv:2208.08911v4

The theorem quantifies over every probability measure $\mu$ on $(0,\infty)$ but uses $d(\eta_1\circ\mu)/d(\eta_1\circ\alpha)$ . By (2.2) and (2.7), $\eta_1\circ\alpha=\beta=\eta_1^2m$ , a non-atomic measure. For the permitted initial law $\mu=\delta_x$ , one has $\eta_1\circ\mu=\delta_x$ , so $\delta_x\not\ll\beta$ and the displayed derivative does not exist. The preceding assertion that every $\mu$ satisfies $\mu\ll\alpha$ is therefore false. A substantive repair must either restrict $\mu$ to laws whose $\eta_1$ -tilt has an $L^2(\beta)$ density or state a bound in terms of a positive-time regularization; the current universal statement cannot be read literally.

Full paper, version 4 ↗

Theorem 1.1Not able to verify

Uniform exponential convergence is not established by the supplied proof

Pages 4-5 and 11-15 · Theorem 1.1 and its proof · arXiv:2208.08911v4

The implication from an entrance boundary at $\infty$ to a bound uniform over all initial laws depends on Proposition 2.2 and on the passage from the $L^1(\beta)$ estimate (3.1) to (3.8). Proposition 2.2 is not proved, and an $L^1(\beta)$ norm bound on a function does not control its integral against an arbitrary, possibly singular, probability measure $\eta_1\circ\mu$ . No independent argument covering the claimed oscillatory exit-boundary regime is supplied. These defects do not disprove the theorem, but they leave the advertised uniform implication unverified.

Champagnat-Villemonais, earlier criteria for uniform convergence ↗

Proposition 1.3Not able to verify

The uniform conditional empirical-measure rate remains unverified

Pages 5-6 and 17-19 · Proposition 1.3 and its proof · arXiv:2208.08911v4

The $G/t$ estimate is derived from the strong-ergodicity estimate (5.3), which is exactly the unproved conclusion of Proposition 2.2. Since the claimed constants must be uniform in $x$ , ordinary ergodicity of the $Q$ -process does not fill this dependency. No counterexample was established, but the only supplied proof does not verify the proposition.

Full paper, version 4 ↗

02Proofs4 reported findingsContains incorrect or incomplete proofs

The Foster-Lyapunov proof uses a function that is not even continuously differentiable at its splice point and then treats a non-invariant truncated interval as a state space. The proof of Theorem 1.1 also applies an $L^1(\beta)$ function estimate to arbitrary singular initial laws, and the proof of Theorem 1.2 never repairs the missing Radon-Nikodym derivative.

Proposition 2.2Incorrect as written

The Foster-Lyapunov construction does not prove strong ergodicity

Pages 10-11 · Equations (2.12)-(2.18) · arXiv:2208.08911v4

The proposed Lyapunov function equals $1$ on $(0,b]$ and $\overline L(x)=1+\int_b^x e^{\widetilde Q(y)}\int_y^\infty e^{-\widetilde Q(z)}\,dz\,dy$ on $[b,\infty)$ . Its left derivative at $b$ is $0$ , whereas its right derivative is $e^{\widetilde Q(b)}\int_b^\infty e^{-\widetilde Q(z)}\,dz>0$ ; hence the printed claim $L\in C^2((0,\infty))$ and the global generator calculation are false. Independently, inaccessibility of the boundary point $0$ does not imply that a diffusion started above $\varepsilon$ never enters $(0,\varepsilon)$ . Thus $E_\varepsilon=[\varepsilon,\infty)$ is not an invariant state space, and the constants $D_\varepsilon$ in (2.17) cannot be replaced by a uniform $D$ merely because $\varepsilon$ is arbitrary. Repair classification: No repair supplied; a valid global Lyapunov or minorization argument with constants uniform near $0$ is required.

Full paper, version 4 ↗

Proof of Theorem 1.1Incomplete as written

Equation (3.8) does not follow from the $L^1(\beta)$ estimate

Pages 12-15 · Equations (3.1) and (3.8)-(3.10) · arXiv:2208.08911v4

Equation (3.1) controls $\|\widetilde P_t f-\beta(f)\|_{L^1(\beta)}$ . Equation (3.8) instead integrates $\widetilde P_t(g/\eta_1)-\beta(g/\eta_1)$ against $\eta_1\circ\mu$ and claims a bound uniform in every initial law $\mu$ . An $L^1(\beta)$ estimate gives no such control when $\eta_1\circ\mu$ is singular; taking $\mu=\delta_x$ makes the mismatch explicit. Strong total-variation ergodicity would control bounded test functions, but $g/\eta_1$ need not be bounded. Repair classification: No repair supplied; a pointwise or appropriately weighted uniform bound is needed.

Full paper, version 4 ↗

Proof of Theorem 1.2Incorrect as written

Positive-time smoothing does not justify the printed initial-density norm

Pages 15-17 · beginning of Section 4 and Equations (4.2)-(4.3) · arXiv:2208.08911v4

The proof says that an infinite density norm makes the theorem trivial, but for a singular initial law the Radon-Nikodym derivative is nonexistent, not an $L^2$ function of infinite norm. The later identity $\eta_1\circ\phi_t(\mu)=(\eta_1\circ\mu)\widetilde P_t$ may regularize the law at positive time, yet the final displayed estimate again bounds the regularized norm by $\|d(\eta_1\circ\mu)/d\beta-1\|_2e^{-\gamma t}$ , which is unavailable for the stated $\mu$ . Repair classification: No repair supplied; a finite norm at a specified positive time could yield a different estimate with a shifted exponential factor, but it is not the theorem printed.

Full paper, version 4 ↗

Proof of Proposition 1.3Incomplete as written

The bridge estimate depends on the unproved uniform $Q$ -process bound

Pages 17-19 · Equations (5.3)-(5.4) · arXiv:2208.08911v4

The calculation after (5.3) is algebraically consistent once a uniform exponential bound for the $Q$ -process is available, but (5.3) is taken directly from Proposition 2.2. Because the proof of that proposition fails at (2.12)-(2.18), the essential input to the conditional empirical-measure estimate is missing. Repair classification: No repair supplied beyond repairing Proposition 2.2.

Full paper, version 4 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2208.08911v4
Authors listed: Guoman He, Hanjun 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 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. 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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)

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

and subsequently applied at

t=0

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Uniform convergence of conditional distributions for one-dimensional diffusion processes

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The displayed bound is undefined for allowed initial distributions

Uniform exponential convergence is not established by the supplied proof

The uniform conditional empirical-measure rate remains unverified

The Foster-Lyapunov construction does not prove strong ergodicity

Equation (3.8) does not follow from the L1(β)L^1(\beta)L1(β) estimate

Positive-time smoothing does not justify the printed initial-density norm

The bridge estimate depends on the unproved uniform QQQ-process bound

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Equation (3.8) does not follow from the $L^1(\beta)$ estimate

The bridge estimate depends on the unproved uniform $Q$ -process bound