{"id":"fc06d066-bd9f-4864-a64f-262166a1abf7","arxiv_id":"2509.01519","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"If the deterministic skeleton converges to zero and small jumps vanish, a symmetric pure-jump SDDE is weakly irreducible to zero.","lead":"This paper proves a criterion for weak irreducibility (accessibility) of stochastic delay differential equations driven by pure jump Levy noise, and applies it to equations with weakly dissipative coefficients. The result gives a concise condition under which solutions can reach any neighborhood of the zero path with positive probability, a stepping stone for uniqueness of invariant measures.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.1's verification of (A1-2) rests on a nonstandard stochastic-Gronwall application; (3.12) needs independent justification.","rationale":"The reader's weakest-assumption diagnosis (A1-2) is correct: the entire application hinges on the small-jump truncation converging to the deterministic path, and the proof of that convergence is the least secure part. The central Theorem 2.1 is a clean conditional result, but Proposition 3.1's stochastic Gronwall step is too terse and potentially misapplied. This is not a disagreement with the reader's verdict; it reinforces CONDITIONAL. The q1=2 division is a separate but minor technical error. Since the main idea is sound and the gaps are checkable, no verdict change is needed.","tokens_in":8813,"tokens_out":23626,"duration_ms":246190,"concrete_test":"Reproduce the derivation of (3.12) from (3.11) by explicitly stating Lemma 3.7 of [24] and checking its hypotheses. In particular, verify whether the lemma applies to an inequality whose RHS contains a martingale with integrand depending on Y. If the lemma requires a Y dA term, re-derive (3.12) by hand using BDG and Young; if the constant or epsilon-dependence differs from the claimed p/(p-q) bound, then A1-2 is not proven.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem is conditional on Assumption (A1-2), and Proposition 3.1 is the only verification. In its proof, after Ito's formula the paper reaches (3.11): Y_t := e^{2rt}|X^{eps,xi}(t)-X^xi(t)|^2 ≤ 2M_t + I_t, with M_t = ∫∫ e^{2rs}⟨D_s,z⟩\\tilde N and I_t = ∫∫ e^{2rs}|z|^2 N. The right side has no Y-term except inside the martingale integrand; the quadratic variation of M depends on Y, [M]_t ≤ Y^*_t I_t. The paper cites Lemma 3.7 of [24] and immediately asserts (3.12), a bound on E[sup Y]^q by a power of ∫ e^{2rs}|z|^2ν dz ds. Standard stochastic Gronwall requires a term ∫_0^t Y_{s-} dA_s, which is absent. A valid bound would need an additional argument (e.g., solving the quadratic inequality via BDG and Young). If the cited lemma does not cover this self-referential case, (A1-2) is not established and the weakly dissipative application collapses. Separately, (3.6) divides by q1r-2r without excluding q1=2; this is fixable but further shows the proof is incomplete.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies weak irreducibility (accessibility to zero) for stochastic delay differential equations driven by additive pure-jump Lévy noise on the weighted path space D_r. The main result, Theorem 2.1, gives sufficient conditions: symmetric Lévy measure, unique global solutions to the original/truncated/deterministic equations, deterministic solutions attracted to zero (A1-1), and convergence of small-jump truncations to the untruncated solution in probability (A1-2). The proof couples the original solution with the truncated solution up to the first big jump, using independence of the big-jump process and positivity of the probability of no big jump. Section 3 claims to verify these assumptions for weakly dissipative coefficients satisfying a one-sided Lipschitz-type condition (H2), with an application to a nonlinear delay example.","tokens_in":9115,"tokens_out":24247,"duration_ms":283904,"significance":"If the main theorem and the application were both correct, the paper would provide a short, appealing sufficient condition for weak irreducibility of SDDEs driven by pure jump noise, and would relax condition (A1-3) from earlier work. The coupling idea in Theorem 2.1 is elegant and essentially self-contained: it does not rely on the cited framework for the key step, and it is not circular, since A1-1 and A1-2 are genuine sufficient conditions rather than restatements of weak irreducibility. However, the application in Proposition 3.1 has serious gaps. In particular, the proof of (A1-1) is invalid as written and the stated proposition admits a simple counterexample; a second load-bearing estimate, (3.12), is not justified by the cited stochastic Gronwall lemma. These issues materially affect the advertised application, so the paper cannot be accepted in its current form.","major_comments":[{"comment":"The verification of (A1-1) is incorrect. The stated bound ||X^ξ_t||_r ≤ e^{-rt} sup_{0≤s≤t}|X(s)| + e^{-rt}||ξ||_r is false; the correct bound is e^{-rt} sup_{0≤s≤t} e^{rs}|X(s)| + e^{-rt}||ξ||_r. Consequently, the argument only proves sup_t |X(t)| < ∞, which does not imply ||X^ξ_t||_r → 0. The issue is load-bearing: take f(φ) = -φ(0)+1, r ≤ 1, and ν = 0. Then (H1) and (H2) hold with λ̄1=0, λ̄2=2, K̄1=K̄2=0, and the stated inequalities are satisfied, but the deterministic solution starting from the constant path ξ≡1 is X(t)=1 for all t, so the family is not weakly irreducible to zero. Thus Proposition 3.1 is false as stated. The proposition needs additional hypotheses (e.g., f(0)=0 and strict dissipativity) and the example, which has f(0)=1, must be reconsidered.","section":"Section 3, proof of Proposition 3.1, paragraph after Eq. (3.7)"},{"comment":"The stochastic Gronwall step is not justified. After dropping negative terms, (3.11) has the form Y_t ≤ 2M_t + I_t, with no Y-term in the drift; the cited Lemma 3.7 of [24] is applied without verifying its hypotheses. Controlling E[sup Y^q] requires an additional argument: by BDG, E[sup|M|^q] is controlled by E[[M]_t^{q/2}], and [M]_t ≤ Y^*_t I_t, leading to a quadratic inequality in E[(Y^*)^q]. This is not what is written. As it stands, (3.12) is unsupported, and since (A1-2) is the only verification of a key input to Theorem 2.1, this is a serious gap.","section":"Eq. (3.11)–(3.12), proof of (A1-2)"},{"comment":"The estimate divides by (q1 r - 2r) without excluding q1 r = 2r. Since q1 > 0 is arbitrary, the case q1 = 2 is allowed by the assumptions, and the displayed formula is undefined there. This is repairable by treating q1 = 2 separately, but it shows the proof of the a priori bound is incomplete.","section":"Eq. (3.6)"}],"minor_comments":[{"comment":"The word 'irreduciblility' is misspelled in several places. Please proofread throughout.","section":"Abstract/Introduction"},{"comment":"The integration limits in the first line, e.g. ∫_{-s}^{-∞}, should be written as ∫_{-∞}^{-s}; the current notation is confusing.","section":"Eq. (3.5)"},{"comment":"In the stochastic integrals the integrand should be the left limit X^{ε,ξ}(s-) - X^ξ(s-), not X^{ε,ξ}(s) - X^ξ(s). The intended estimate still works with left limits, but the notation should be corrected.","section":"Eq. (3.8) and (3.11)"},{"comment":"In the chain of inequalities after Eq. (2.5), on the event τ_ε > T one has x^ξ_T = X^{ε,ξ}_T, so the displayed expression reduces to ||X^{ε,ξ}_T - X^ξ_T + X^ξ_T||; the intermediate form is harmless but should be clarified to avoid confusion.","section":"Theorem 2.1 proof"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is sound, but Proposition 3.1 is false as stated; the counterexample f(φ) = -φ(0)+1 with ν=0 satisfies all hypotheses and violates the conclusion. This is not a local typo: it requires changing the statement of the application and the example. If the authors cannot repair this by adding the needed dissipativity/nondegeneracy assumptions and reworking the example, the paper would not be publishable in its present form. The stochastic Gronwall gap is also significant but appears repairable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe one thing to know: this paper has a genuinely clean one-big-jump proof of weak irreducibility for SDDEs driven by pure jump noise, and that proof works. The soft spot is the application: Proposition 3.1's verification of assumption (A1-2) leans on a stochastic Gronwall step that is not justified as written. Fix that and the paper is solid; as is, the main theorem is a conditional result but the abstract's stronger claim rests on the gap.\n\nWhat is new here: this appears to be the first weak irreducibility result for infinite-delay SDDEs with pure jump Lévy noise, and the framework allows degenerate symmetric noise. The proof of Theorem 2.1 is exactly right in spirit: a large jump occurs with positive probability before T, the small-jump process is close to the deterministic skeleton, and the skeleton decays to zero. The independence between the small-jump process and the first big-jump time is used correctly. That part deserves credit.\n\nThe problems are in the application. Three specific things:\n\n1. Equation (3.6) divides by (q1 r - 2r) without stating q1 r > 2r (or at least ≠ 2r). If q1 r < 2r the inequality direction flips and the bound becomes negative; the stated assumptions do not rule this out. This is a minor fix, but it has to be in the statement.\n\n2. The step from (3.11) to (3.12) is the real issue. The inequality Y ≤ 2M + I has no drift term in Y, and the martingale quadratic variation is controlled by sup Y times I. That is not the form covered by the usual stochastic Gronwall lemma (Lemma 3.7 in [24]); the paper just asserts it. One can likely get the bound via BDG plus Young, but that argument is not present. Since (3.12) is the only verification of (A1-2), this is load-bearing for Proposition 3.1. The conditional Theorem 2.1 does not depend on it.\n\n3. Minor: the proposition's conditions include the inequalities on λ̄ and K̄, but the example only checks one numerical case; that is fine.\n\nBottom line: the core idea is good and the central theorem is probably right. The paper deserves refereeing, but the referee should send it back for a proof of (3.12) and clean-up of (3.6). I would not cite the application until that's fixed, though I'd cite Theorem 2.1 as a conditional tool. You could bring it to reading group to discuss the one-big-jump trick, but the stochastic Gronwall gap will take attention away from the actual proof.","headline":"A clean one-big-jump proof of weak irreducibility for SDDEs with pure jump noise, undermined by an unsupported stochastic Gronwall step in the only application.","tokens_in":9642,"tokens_out":7064,"would_cite":true,"duration_ms":74531,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34K50","60H10","60G51","37A50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that stochastic delay equations driven by pure jump Lévy noise are weakly irreducible to zero: from every initial history, each neighborhood of zero is visited with positive probability.","keywords":["Weak irreducibility","Accessibility to zero","Stochastic delay differential equations","Pure jump Lévy noise","Weakly dissipative coefficients","Degenerate noise","Invariant measures"],"falsifier":"The claim would be refuted by an SDDE that meets Assumption 2.1 yet has some neighborhood of zero that the solution never enters with positive probability. A concrete place to look is f(φ)=aφ(0)+bφ(-r) with symmetric stable-like noise: solve the linear SDDE explicitly via Fourier or Laplace transforms in the path space and compute P(||x_T||_r≤κ) for large T. If this probability is zero for some κ>0, Theorem 2.1's conclusion fails; equivalently, computing the left side of (3.12) for that example decides whether condition (A1-2) actually holds.","tokens_in":8650,"feed_emoji":"🎲","tokens_out":8583,"duration_ms":104708,"temperature":0.7,"pith_summary":"The paper proves a reachability property, weak irreducibility to zero, for stochastic delay differential equations driven by additive pure jump Lévy noise: from any initial history segment, the solution has positive probability of passing through any small neighborhood of the zero history. The proof is a short coupling: wait for a single jump larger than a cutoff; until that jump occurs, the full noise process equals the small-jump-truncated noise, so the solution agrees with a truncated equation that already tracks the deterministic solution. The authors extract two checkable assumptions—the deterministic solution decays to zero, and removing small jumps changes the solution negligibly on finite time horizons—and verify them for weakly dissipative coefficients. This matters because weak irreducibility, paired with a smoothing or contractivity property, is the standard route to uniqueness of invariant measures for Markov processes on infinite-dimensional path spaces. The paper also allows the driving noise to be degenerate, which earlier irreducibility results for pure-jump systems did not.","feed_headline":"One big jump makes delay equations weakly irreducible","feed_subtitle":"Pure jump Lévy noise, even degenerate, lets delay systems reach any neighborhood of zero with positive probability.","key_machinery":"The load-bearing device is the one-big-jump coupling. One compares the full equation to the truncated equation containing only jumps of size ≤ ε; before the first jump of size > ε, the two paths are identical. The first large-jump time is exponentially distributed and independent of the truncated solution, so the probability of ending in a target neighborhood factorizes into: the deterministic solution near zero (A1-1), the truncated solution near the deterministic one with probability at least 1/2 (A1-2), and the large jump not having happened by time T. Symmetry of the Lévy measure is used so that small jumps enter as a compensated martingale and the large-jump process is an independent Po","core_discovery":"On the Banach space D_r of history paths with exponentially weighted sup-norm, consider d x(t) = f(x_t) dt + dL(t), where L is a pure jump Lévy process with symmetric intensity measure. Theorem 2.1 states: if unique global solutions exist, the deterministic limit dX = f(X_t)dt satisfies ||X_t^ξ||_r → 0, and the small-jump truncations X^{ε,ξ} converge to X^ξ in probability in the path norm for each fixed time, then the family {x^ξ} is weakly irreducible to zero. The proof fixes a neighborhood of zero, chooses T so the deterministic path is inside half the neighborhood, chooses ε so the truncated path stays close to it with probability at least 1/2, and then waits for the first jump of size >","pith_inferences":["Beyond the paper: a natural extension would replace convergence of truncations in probability by a quantitative bound—for instance an L^q estimate as in (3.12)—and use it to estimate the actual hitting probability, giving rates rather than mere positivity.","Beyond the paper: the one-big-jump mechanism does not obviously need symmetry of the whole Lévy measure; any decomposition into a compensated small-jump part and an independent large-jump Poisson part with the same pathwise agreement would do, so the result may carry over to asymmetric measures with an appropriate centering term.","Beyond the paper: for applications to ergodicity, the missing half is a smoothing or eventual-continuity property; the paper proves only the reachability ingredient, so the next test is whether these same weakly dissipative SDDE semigroups are eventually continuous.","Beyond the paper: a stress test for the assumptions is the linear case f(φ)=aφ(0)+bφ(-r), where explicit Laplace-transform formulas for P(||x_T||_r≤κ) could show whether A1-2 is also necessary or merely sufficient."],"forward_implications":["Weakly dissipative SDDEs driven by additive pure jump Lévy noise are weakly irreducible to zero whenever the two inequalities in Proposition 3.1 hold; the example equation (3.13) is covered.","Degenerate symmetric pure jump noise suffices, so delay equations with noise acting through a one-dimensional component or a single direction inherit accessibility to zero.","Combined with any uniqueness criterion for invariant measures—asymptotic strong Feller, e-property, or eventual continuity—weak irreducibility yields at most one invariant probability measure for these path-dependent systems.","The criterion in Theorem 2.1 is model-independent: any coefficient class for which one can prove deterministic decay and small-jump truncation convergence in the path norm automatically gets weak irreducibility.","Releasing condition (A1-3) from the earlier criterion [21] broadens the applicability of accessibility results from SPDEs to delay equations whose coefficients need not satisfy that stronger small-jump condition."],"supporting_citations":[{"why":"Supplies the prior accessibility criterion for SPDEs with condition (A1-3) that this paper releases and generalizes to SDDEs.","marker":"[21]"},{"why":"Supplies the stochastic Gronwall inequality used to prove the small-jump truncation convergence required by condition (A1-2).","marker":"[24]"},{"why":"Supplies the semimartingale existence and uniqueness theory invoked for global solutions of the truncated and deterministic equations.","marker":"[14]"},{"why":"Supplies the interlacing technique used to construct the global solution of the original pure-jump equation from the truncated equation.","marker":"[2]"},{"why":"Defines the Lévy process and Poisson random measure integral representation underlying the noise.","marker":"[15]"},{"why":"Provides a global-solution result for functional SDEs used in verifying unique global solutions under hypotheses (H1)-(H2).","marker":"[4]"},{"why":"Supplies the asymptotic strong Feller smoothing notion that motivates why weak irreducibility matters for ergodicity.","marker":"[7]"}],"fun_headline_variants":["Degenerate pure jumps make delay systems weakly irreducible","Short proof: weak irreducibility for delay equations with pure jumps","Delay systems weakly irreducible under degenerate jump noise","Even degenerate jumps ensure weak irreducibility of delay equations","Pure jump noise: delay systems reach zero without ellipticity"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that small jumps are negligible in the path norm—deleting all jumps below a cutoff changes the history path by an arbitrarily small amount in probability at each fixed time; if that fails, the one-big-jump coupling cannot confine the solution to a neighborhood of zero.","fun_headline_variants_meta":{"raw":{"variants":["Degenerate pure jumps make delay systems weakly irreducible","Short proof: weak irreducibility for delay equations with pure jumps","Delay systems weakly irreducible under degenerate jump noise","Even degenerate jumps ensure weak irreducibility of delay equations","Pure jump noise: delay systems reach zero without ellipticity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000252,"raw_usage":{"total_tokens":1352,"prompt_tokens":655,"completion_tokens":697,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":399,"completion_tokens_details":{"reasoning_tokens":619}},"tokens_in":399,"tokens_out":697,"duration_ms":8810,"temperature":1.0,"reasoning_tokens":619,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:29:35.990968+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The claim would be refuted by an SDDE that meets Assumption 2.1 yet has some neighborhood of zero that the solution never enters with positive probability. A concrete place to look is f(φ)=aφ(0)+bφ(-r) with symmetric stable-like noise: solve the linear SDDE explicitly via Fourier or Laplace transforms in the path space and compute P(||x_T||_r≤κ) for large T. If this probability is zero for some κ>0, Theorem 2.1's conclusion fails; equivalently, computing the left side of (3.12) for that example decides whether condition (A1-2) actually holds.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the prior accessibility criterion for SPDEs with condition (A1-3) that this paper releases and generalizes to SDDEs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the stochastic Gronwall inequality used to prove the small-jump truncation convergence required by condition (A1-2)."},{"cited_title":"Protter, Stochastic Integration and Differential Equations","cited_arxiv_id":null,"evidence_quote":"Supplies the semimartingale existence and uniqueness theory invoked for global solutions of the truncated and deterministic equations."},{"cited_title":"Brze´ zniak, W","cited_arxiv_id":null,"evidence_quote":"Supplies the interlacing technique used to construct the global solution of the original pure-jump equation from the truncated equation."},{"cited_title":"Peszat, J","cited_arxiv_id":null,"evidence_quote":"Defines the Lévy process and Poisson random measure integral representation underlying the noise."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides a global-solution result for functional SDEs used in verifying unique global solutions under hypotheses (H1)-(H2)."},{"cited_title":"Hairer, J.C","cited_arxiv_id":null,"evidence_quote":"Supplies the asymptotic strong Feller smoothing notion that motivates why weak irreducibility matters for ergodicity."}],"review_version":1}