arXiv:2607.27932v2

Freidlin-Wentzell collision-laws between self-stabilizing diffusions

Jean-François Jabir, Julian Tugaut

math.PR60H1060J6060K3537A50

Paper metadata

Original submission: July 30, 2026
Latest revision: August 1, 2026 · version 2
Claimed MSC: 60H10, 60J60, 60K35, 37A50
arXiv categories: math.PR
Paper license: arXiv non-exclusive distribution license ↗
Comments: The present paper features a completely revised and augmented version of the unpublished paper: arXiv:2206.04542

Abstract

The present work investigates the asymptotic behaviours at the zero-noise limit of the first near collision-time and first near collision-location between a pair of independent $d$ -dimensional Brownian-driven self-stabilizing (McKean-Vlasov type) diffusions. These asymptotic are considered in a peculiar setting where the systems evolve in a bi-stable landscape and collisions are only triggered by the combined action of the Brownian noises. As the Brownian perturbations fade away, we show that the near collision-time increases at an explicit exponential rate and that related collision-locations persist in specific regions of the space. These results are mainly derived by tailoring classical Freidlin-Wentzell's exit-time (and exit-location) estimates into collision estimates. Similar asymptotic are established for related mean-field interacting particle system approximation, and for the one-dimensional case (where true collisions can be examined

AI-generated audit

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 findingsContains unsupported statements

No counterexample to either principal collision law was established. Both remain unverified under the printed assumptions, however. The proof applies the paper's bounded-domain, globally-Lipschitz Freidlin--Wentzell theorem to unbounded domains and polynomial drifts, and its exponentially long coupling uses a local quadratic stability property that does not follow from asymptotic stability. The particle theorem also requires a population threshold depending on the collision radius, while its statement quantifies over one fixed sufficiently large population before taking the radius to zero.

Theorem 1.5Not able to verify

Collision law for the two self-stabilizing diffusions

Pages 9 and 13–28 · Theorem 1.5 and Sections 2–3 · arXiv:2607.27932v2

The action $H_0$ is the natural candidate for the iterated small-radius and small-noise collision cost, and the finite-cover argument would transfer fixed-center hitting laws to the first collision. Two indispensable inputs are not established under Assumptions (A), though. First, Lemma 2.2 invokes Theorem 1.4 outside the theorem's printed bounded-domain and globally-Lipschitz hypotheses. Second, Proposition 3.2 assumes local quadratic attraction of $-U$ at each well, whereas (A)-(ii) gives only Lyapunov asymptotic stability. Since these inputs control the process for times of order $\exp(2H/\sigma^2)$ , the displayed double limit and persistence conclusion are not verified by the present proof. This is not a claim that the theorem is false.

Theorem 1.6Not able to verify

Collision law for a fixed sufficiently large particle population

Pages 10 and 29–35 · Theorem 1.6 and Section 4 · arXiv:2607.27932v2

This theorem inherits the unresolved Freidlin--Wentzell and local-stability inputs from Theorem 1.5. It also states the iterated limit for every fixed $N$ above one unspecified threshold. Proposition 4.2 instead produces $N_{\varepsilon,\kappa}$ depending on the collision radius and coupling tolerance; the transfer to a radius- $\varepsilon$ hitting event requires the tolerance to be smaller than $\varepsilon$ . No uniform bound on $N_{\varepsilon,\kappa}$ as $\varepsilon\downarrow0$ is proved. The argument therefore verifies, at most, a formulation in which $N$ is chosen after fixing $\varepsilon$ , not the printed quantifier order.

02Proofs6 reported findingsContains incorrect or incomplete proofs

The finite-cover reductions and many coupling estimates are internally consistent after local notation corrections, but three central obligations remain: extending the exit theorem to the unbounded polynomial-drift setting actually used, justifying exponentially long localization from the stated stability assumption, and making the particle-rank choice uniform in the outer radius limit. The printed definition of the positive small-radius threshold also uses an infimum where its own explanation requires a supremal interval.

Lemma 2.2 and Lemma 2.4Incomplete as written

The recalled exit theorem is applied outside two explicit hypotheses

Pages 8 and 13–17 · Theorem 1.4 and Lemmas 2.2–2.4 · arXiv:2607.27932v2

Theorem 1.4 in the paper assumes an open bounded exit domain and globally Lipschitz drift and diffusion. Lemma 2.2 applies it to $G=(D^1_{\lambda,\varepsilon}\times D^2_{\lambda,\varepsilon})^c$ and its modified versions. These domains are generally unbounded; for the one-dimensional contractive drift $-\Psi_i(x)=-(x-\lambda_i)$ , each complement $(D^i_{\lambda,\varepsilon})^c$ is already an unbounded half-line when the target ball misses $\lambda_i$ . Moreover, the main drifts $W_{\lambda_i}=U+F(\,·-\lambda_i)$ have the polynomial growth allowed by (A)-(i) and (A)-(iii), not global Lipschitz continuity. No extension theorem or truncation-and-outer-boundary argument is supplied. Downstream dependency: Lemma 2.4, Propositions 2.5, 3.3–3.4, and both main theorems. A repair must prove an unbounded-domain exit law under the paper's coercive polynomial hypotheses or truncate the domains and show uniformly that every artificial outer boundary has strictly larger action.

Proposition 3.2, Step 4Incomplete as written

Asymptotic stability is used as if it implied quadratic local coercivity

Pages 26–27 · Step 4 of Proposition 3.2 · arXiv:2607.27932v2

The proof asserts that the Jacobian of $-U$ at $\lambda_1$ is negative definite and hence that, for some $\rho,r>0$ , $B(\lambda_1;r)$ lies in $S_\rho=\{x:(x-\lambda_1)\cdot U(x)\geq\rho\lVert x-\lambda_1\rVert^2\}.$ Assumption (A)-(ii) states only Lyapunov asymptotic stability. That does not imply hyperbolicity or this quadratic bound: the scalar flow $\dot x=-x^3$ is asymptotically stable at zero but satisfies $xU(x)=x^4$ , which cannot dominate $\rho x^2$ near zero. The other assumptions do not add the asserted property for $U$ itself. This bound is then used to keep $\mathbb E\lVert X_t-\lambda_1\rVert^2$ small up to exponentially large times. A verified sufficient repair is to assume explicitly that $(x-\lambda_i)\cdot U(x)\geq\rho_i\lVert x-\lambda_i\rVert^2$ in a neighborhood of each well, or replace Step 4 by a localization argument valid for nonhyperbolic stable equilibria.

Theorem 1.6 and Proposition 4.2Incomplete as written

The population threshold is not uniform in the radius limit

Pages 10 and 29–35 · Theorem 1.6, Proposition 4.2, and Section 4.2 · arXiv:2607.27932v2

Theorem 1.6 first fixes a sufficiently large $N$ and then takes $\varepsilon\downarrow0$ . Proposition 4.2 chooses $N_{\varepsilon,\kappa}$ after $\varepsilon$ and $\kappa$ , and the bracketing of radius- $\varepsilon$ hitting times requires $\kappa<\varepsilon$ . The proof gives no bound on $\sup_{0<\varepsilon<\varepsilon_*}N_{\varepsilon,\kappa(\varepsilon)}$ . Repair classification: unresolved under the printed theorem. Either establish such a uniform bound, or change the conclusion to state the fixed- $\varepsilon$ law with $N\geq N(\varepsilon)$ and specify a joint order of limits.

Equations defining $\varepsilon_c$Typo

An infimum cannot define the advertised largest small-radius interval

Pages 17 and 27 · definitions preceding Propositions 2.5 and 3.4 · arXiv:2607.27932v2

Both definitions print $\varepsilon_c$ as the infimum of radii at which the desired strict comparison holds, then call it the largest radius and use the comparison for every $0<\varepsilon<\varepsilon_c$ . If the comparison holds for all sufficiently small radii, the printed infimum is zero. The definition dictated by the following prose is $\varepsilon_c=\sup\{r>0:\text{the comparison holds for every }\varepsilon\in(0,r)\}.$ This symbol-level correction does not change the intended results, but the proof must still verify that this set contains a positive interval.

Assumptions and coupling notationTypo

Four local symbols conflict with their immediately stated meaning

Pages 5, 22, and 24 · (A)-(i), (A)-(ii), and Proposition 3.2 · arXiv:2607.27932v2

The strong-convexity inequality in (A)-(i) is printed with $\leq$ although the subsequent deductions require $\xi\cdot\nabla U(x)\xi\geq C\lVert\xi\rVert^2$ ; (A)-(ii) writes $B(\lambda_1;\varepsilon)$ for both wells instead of $B(\lambda_i;\varepsilon)$ . In Proposition 3.2, $S_\kappa^{(1)}$ and the force estimate use $\mathbb E\lVert X_{T_\kappa}-\lambda_1\rVert^2$ where their dependence on the running time and every later line require $\mathbb E\lVert X_t-\lambda_1\rVert^2$ . These replacements are uniquely determined by the surrounding logic and are treated as typos, not adverse statement findings.

Proposition 4.2, Step 5.2.3Incorrect as written

A product probability is used despite interacting particles

Page 33 · Step 5.2.3 of Proposition 4.2 · arXiv:2607.27932v2

The proof bounds the probability that at least $cN$ tagged couplings fail by a binomial sum involving $\nu(\sigma)^\ell$ . The failure events are not independent because the $X^{i,N}$ interact through their empirical measure, so the product power is unjustified. Repair classification: verified repair. Exchangeability and Markov's inequality give $\mathbb P\!\left(\frac1N\sum_{i=1}^N\mathbf1_{A_i}\geq c\right)\leq\frac{\mathbb E[N^{-1}\sum_i\mathbf1_{A_i}]}{c}=\frac{\mathbb P(A_1)}{c}\longrightarrow0,$ which supplies exactly the high-probability conclusion needed in the following step.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2607.27932v2
Authors listed: Jean-François Jabir, Julian Tugaut
Audit date: August 18, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v2
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

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. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. 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

and

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

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

Freidlin-Wentzell collision-laws between self-stabilizing diffusions

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Collision law for the two self-stabilizing diffusions

Collision law for a fixed sufficiently large particle population

The recalled exit theorem is applied outside two explicit hypotheses

Asymptotic stability is used as if it implied quadratic local coercivity

The population threshold is not uniform in the radius limit

An infimum cannot define the advertised largest small-radius interval

Four local symbols conflict with their immediately stated meaning

A product probability is used despite interacting particles

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match