{"id":"e18357da-449a-4b10-834c-a12184036bdd","arxiv_id":"2607.07628","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper defines enhanced probabilistic well-posedness by imposing stability at the origin and reinterprets recent 'beyond variance blowup' results for dispersive PDEs as probabilistic ill-posedness.","lead":"This paper introduces a stricter notion of when a PDE with random initial data is well-posed, requiring stability as the random input amplitude shrinks to zero. It reinterprets recent 'beyond variance blowup' results as evidence of probabilistic ill-posedness, clarifying when randomness cannot rescue a deterministically ill-posed equation.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. The logical structure—Definition 1.8, Proposition 2.1, and its application to the companion-paper convergence results—is internally sound. The main limitation is conceptual, not mathematical.","rationale":"The reader correctly identified that the paper's significance depends on the community adopting Definition 1.8, and correctly assessed the self-citation as non-circular. The paper is a well-executed conceptual note: it introduces a natural definition, applies it coherently to re-interpret existing results, and the mathematical logic is sound. The ACCEPT verdict with HIGH confidence is appropriate. The correctness risk is listed as 'unknown' by the reader, which is reasonable given that the key inputs come from preprint companion papers [59, 58]; however, the current paper's own logical contribution (Definition 1.8 + Proposition 2.1 + the deduction) is internally consistent and does not contain identifiable errors. No verdict adjustment is needed.","tokens_in":25614,"tokens_out":6506,"duration_ms":336910,"concrete_test":"Independently verify the portmanteau theorem step in equation (2.18) for the NLW case: confirm that the random variables A_N = ||u_N||_{C_{T_ω} H^{-1/2-ε}} and A = ||u||_{C_{T_ω} H^{-1/2-ε}} (defined on the Skorokhod-represented probability space) satisfy lim inf E[1_{A_N > λ}] ≥ E[1_{A > λ}] > 0, and that this inequality transfers back to the original probability space to yield (2.1). In particular, check that the random time interval [0, T_ω] for the limiting SPDE is a.s. positive and that the solutions u_N exist on this interval for all sufficiently large N.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central argument is: (1) define enhanced probabilistic well-posedness with continuity at the origin (iii'); (2) show that 'beyond variance blowup' results from [59, 58] violate (iii') via Proposition 2.1; (3) conclude ill-posedness. I examined each link.\n\nProposition 2.1 bridges between almost-sure convergence (what (iii') requires) and convergence in law to a non-trivial limit (what the companion papers prove). The logic is: if (iii') held, u_N → 0 a.s. on [0,T_ω], hence in probability, hence in law. But the companion papers show u_N → u ≠ 0 in law. By uniqueness of weak limits, this is a contradiction. The portmanteau theorem applications in (2.9) and (2.18) correctly derive lim sup P(||u_N|| > λ) > 0 from the non-triviality of the stochastic limit. This is sound.\n\nFor the NLW case (Section 2.2), the convergence is on a random interval [0, T_ω] where T_ω is the existence time for the limiting SPDE (2.17), which may differ from the T_ω in Definition 1.8(iii') (which comes from Part (i)). However, since both are a.s. positive, one can take the minimum, and the contradiction persists on [0, min(T_ω, T_ω')]. The paper acknowledges this subtlety ('we need to proceed with care') and handles it correctly.\n\nThe self-citation of [59, 58] is non-circular: those papers provide convergence-in-law inputs; this paper provides the definition and the logical deduction. The definition (iii') is a natural analogue of deterministic Hadamard continuity at the origin, not reverse-engineered to produce ill-posedness.\n\nThe reader's identified concern—that community adoption of Definition 1.8 is uncertain—is valid but is a sociological observation, not a mathematical flaw. The companion papers [59, 58] are preprints, introducing a dependency on their correctness, but this is standard and the current paper's logic is sound conditional on those inputs.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This note introduces an enhanced notion of probabilistic well-posedness for dispersive PDEs with random initial data by imposing a continuity-at-the-origin condition (Definition 1.8, condition (iii')). The authors then re-interpret their recent 'beyond variance blowup' results for BBM and quadratic NLW as probabilistic ill-posedness results: the convergence in law of solutions with renormalized vanishing data to non-trivial stochastic limits violates condition (iii'), yielding Theorems 2.4 and 2.5. A third contribution (Section 3) interprets variance blowup itself as 'mild probabilistic ill-posedness' by analogy with the failure of C^k-smoothness of the solution map in the deterministic setting. The logical bridge between convergence in law (what the companion papers prove) and almost-sure convergence (what (iii') requires) is provided by Proposition 2.1 via Egoroff's theorem and the portmanteau theorem.","tokens_in":26266,"tokens_out":1292,"duration_ms":234210,"significance":"The paper provides a clean conceptual framework that unifies several recent 'beyond variance blowup' phenomena under the umbrella of probabilistic ill-posedness, directly analogous to Hadamard's classical criterion. The definition (iii') is natural and the deduction is mathematically sound. The application to BBM (Theorem 2.4) and quadratic NLW (Theorem 2.5) follows logically from the convergence-in-law results in [59, 58]. The extension to stochastic PDEs (Section 2.3) and the interpretation of variance blowup as mild ill-posedness (Section 3) add conceptual value. The self-citation of [59, 58] is non-circular: those papers provide the convergence-in-law inputs, while this paper supplies the definition and the logical deduction. The framework yields falsifiable predictions (e.g., Remark 2.7 on quadratic NLS for 1/4 < α ≤ 1/2).","major_comments":[{"comment":"In Section 2.2 (quadratic NLW), the convergence of u_N to the limiting SPDE solution u holds on a random time interval [0, T_ω] coming from the Skorokhod representation on a new probability space, while condition (iii') in Definition 1.8 requires convergence on [0, T_ω] where T_ω is the local existence time from Part (i) on the original probability space. The paper acknowledges this subtlety ('we need to proceed with care') and argues that since both times are a.s. positive, one can take the minimum. However, the random variables A_N and A defined after (2.17) are defined on the new probability space (post-Skorokhod), while Proposition 2.1 and condition (2.1) are stated in terms of probabilities on the original space. The portmanteau theorem application in (2.18) then mixes these two probability spaces. The authors should clarify explicitly how the convergence-in-law on the original空间,经由","section":null},{"comment":"Proposition 2.1 is stated with a fixed time interval [0, T] where T = T_ω > 0 is the random local existence time from Part (i). However, in the application to NLW (Section 2.2), the convergence from [58] holds on [0, T_ω] where T_ω is the existence time for the limiting SPDE (2.17), which is a different random variable. The paper should state more precisely how Proposition 2.1 applies when the time interval for the convergence result differs from the time interval in condition (iii'). The argument that one takes the minimum is standard but should be made explicit in the proposition or its application.","section":null}],"minor_comments":[{"comment":"In Section 2.2, the symbol T_ω is used both for the random existence time from Definition 1.8(i) and for the random existence time of the limiting SPDE (2.17). Using distinct notation (e.g., T_ω and T'_ω) would make the argument more transparent to the reader.","section":null},{"comment":"In (2.18), the step lim inf E[1_{A_N > λ}] ≥ E[1_{A > λ}] uses Fatou's lemma implicitly. Stating this explicitly would help readers follow the portmanteau theorem machinery.","section":null},{"comment":"Proposition 2.1 states the condition (2.1) with lim sup, but the text immediately after says 'the beyond variance blowup results in fact state that u_N converges in law to a non-trivial solution, verifying the condition (2.1).' It would be clearer to note that convergence in law to a non-trivial limit implies (2.1) via the portmanteau theorem, to tighten the logical flow.","section":null},{"comment":"In Section 3, equation (3.4), the bound sup_N ||d^k/dδ^k u^δ_N|_{δ=0}|| ≤ C_{k,ω} < ∞ is the key condition. The text states that variance blowup implies failure for k=2, but the logical step connecting E[|⟨Ξ_2(P_N u_0), ψ⟩|^2] → ∞ to the failure of sup_N ||Ξ_2(P_N u_0)||_{C_T H^s} < ∞ should be stated for emphasis.","section":null},{"comment":"Reference [58] is listed as a preprint and [59] as an arXiv preprint. The journal status of these at the time of publication should be updated if available.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a short note that largely re-interprets the authors' own recent work [59, 58]. The conceptual contribution is the definition (iii') and the observation that convergence-in-law to a non-trivial limit violates it. This is a legitimate and useful contribution, though the mathematical novelty is modest relative to the companion papers. The handling of the random time interval issue in Section 2.2 is the main point that needs clarification; it appears to be correct but is presented too tersely."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful reading and for recognizing the conceptual contribution of our framework. The two major comments both concern the same subtlety: the interplay between the Skorokhod representation (which lives on a new probability space) and Proposition 2.1 / condition (iii') (which are stated on the original probability space), specifically in the NLW application (Section 2.2). We agree that this point deserves a more explicit explanation in the manuscript and will revise accordingly.","responses":[{"response":"We agree with the referee that this point needs to be made more explicit. The key observation is as follows. The convergence-in-law result from [58] is established on the original probability space: the law of u_N (defined on the original space) converges to the law of u (the limiting SPDE solution). The Skorokhod representation theorem is then used as an intermediate tool: it produces a new probability space on which copies of u_N and u are coupled so that convergence holds almost surely. Crucially, the random variables A_N and A defined after (2.17) are defined on this new space, and their almost-sure convergence implies convergence in distribution of A_N to A. Since A_N (on the new space) has the same distribution as the corresponding norm of u_N (on the original space), the portmanteau theorem yields the inequality in (2.18) as a statement about distributions, and hence about probabilities on the original space. In other words, the Skorokhod representation is used only to deduce convergence in distribution of the norms; the portmanteau theorem then translates this back to a statement about the laws on the original space. We will add a clarifying paragraph in Section 2.2 making this logic explicit, including a precise statement of how the two probability spaces are related and why (2.18) is ultimately a statement about the original space.","revision_made":"yes","referee_comment":"In Section 2.2 (quadratic NLW), the convergence of u_N to the limiting SPDE solution u holds on a random time interval [0, T_ω] coming from the Skorokhod representation on a new probability space, while condition (iii') in Definition 1.8 requires convergence on [0, T_ω] where T_ω is the local existence time from Part (i) on the original probability space. The paper acknowledges this subtlety and argues that since both times are a.s. positive, one can take the minimum. However, the random variables A_N and A defined after (2.17) are defined on the new probability space (post-Skorokhod), while Proposition 2.1 and condition (2.1) are stated in terms of probabilities on the original space. The portmanteau theorem application in (2.18) then mixes these two probability spaces. The authors should clarify explicitly how the convergence-in-law on the original space relates to the almost-sure and,"},{"response":"The referee is correct that the two random times arise from different sources and that the 'take the minimum' argument, while standard, should be stated explicitly. We will add a remark after the application of Proposition 2.1 in Section 2.2 spelling out the following: Let T_ω^(i) denote the random local existence time from Part (i) on the original space, and let T_ω^(SPDE) denote the almost surely positive local existence time for the limiting SPDE (2.17) on the Skorokhod space. Both are a.s. positive, so their minimum T_ω = min(T_ω^(i), T_ω^(SPDE)) is also a.s. positive. The convergence of u_N to u holds on [0, T_ω^(SPDE)], and condition (iii') requires convergence on [0, T_ω^(i)]. On the intersection [0, T_ω], both the convergence and the condition (iii') requirement are satisfied. Since the probability in (2.1) only requires that the norm of u_N exceeds λ on some a.s. positive random time interval, restricting to [0, T_ω] suffices. We will make this explicit in the revised manuscript.","revision_made":"yes","referee_comment":"Proposition 2.1 is stated with a fixed time interval [0, T] where T = T_ω > 0 is the random local existence time from Part (i). However, in the application to NLW (Section 2.2), the convergence from [58] holds on [0, T_ω] where T_ω is the existence time for the limiting SPDE (2.17), which is a different random variable. The paper should state more precisely how Proposition 2.1 applies when the time interval for the convergence result differs from the time interval in condition (iii'). The argument that one takes the minimum is standard but should be made explicit in the proposition or its application."}],"tokens_in":25573,"tokens_out":1045,"duration_ms":107070,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Bottom line: this paper introduces an enhanced definition of probabilistic well-posedness (Definition 1.8) that adds a continuity-at-the-origin condition (iii'), then shows that the authors' recent convergence-in-law results for BBM and quadratic NLW violate this condition, yielding probabilistic ill-posedness. The logic is clean and the conceptual contribution is genuine — it gives a unifying framework for interpreting phenomena that were previously just technical observations about variance divergence and non-trivial stochastic limits in the small-data limit. The paper is well-written and the mathematical content is correct as far as I can tell. The key bridge is Proposition 2.1, which uses Egoroff's theorem to pass from almost-sure convergence (what (iii') demands) to convergence in probability, then derives a contradiction with the non-trivial weak limits from the companion papers via the portmanteau theorem. This is straightforward and sound. The BBM application (Theorem 2.4) is clean. The NLW case (Theorem 2.5) has a minor subtlety — the convergence holds on a random time interval that may differ from the one in Definition 1.8(iii') — but the authors handle this correctly by taking the minimum of the two a.s. positive times. Section 3, interpreting variance blowup as mild ill-posedness via failure of smoothness (analogue of Bourgain's KdV argument), is a nice conceptual parallel and the derivation through the Picard iterate expansion is clear. The self-citation of [59, 58] is non-circular: those papers supply convergence-in-law inputs, this paper supplies the definition and the logical deduction. The main soft spot is conceptual, not mathematical: the significance depends on whether the community adopts condition (iii') as part of the standard definition of probabilistic well-posedness. The authors argue for it by analogy to deterministic Hadamard well-posedness, which is reasonable, but adoption is not guaranteed. This is inherent to a definitional note and does not diminish the correctness of what is proven. The companion papers [59, 58] are preprints, so the ill-posedness conclusions are conditional on those inputs being correct — but that is standard practice and the conditional logic here is sound. This paper is for researchers in probabilistic dispersive PDE theory who want a conceptual framework for understanding where and why probabilistic well-posedness breaks down. It deserves a serious referee, primarily to assess whether the definition is well-motivated and the applications are correctly deduced, not to check deep new estimates.","headline":"Conceptual note reinterpreting variance blowup results as probabilistic ill-posedness; sound logic, worth a referee","tokens_in":26559,"tokens_out":596,"would_cite":false,"duration_ms":66143,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R60","60H15","60H30","35Q35","35L71","35Q53"],"pacs":[],"model":"glm-5.2","headline":"Random data PDEs that look well-posed are actually ill-posed","keywords":["probabilistic well-posedness","ill-posedness","variance blowup","dispersive PDE","random initial data","BBM equation","nonlinear wave equation","continuity at the origin"],"falsifier":"If one can exhibit, for BBM with alpha <= 1/4 or quadratic NLW with beta >= 1/2, that solutions with scaled frequency-truncated data delta_N P_N u_0 do converge to zero in probability (contradicting the convergence-in-law to a non-trivial stochastic limit), then the ill-posedness claim fails.","tokens_in":25706,"feed_emoji":"🎲","tokens_out":1242,"duration_ms":2012529,"temperature":0.7,"pith_summary":"The paper introduces a new criterion for what it means for a dispersive PDE with random initial data to be well-posed: not only must a solution exist and be stable under frequency truncation, but solutions with vanishingly small random data must converge to zero. The authors call this 'continuity at the origin.' They then show that recent 'beyond variance blowup' results for the Benjamin-Bona-Mahony equation and the quadratic nonlinear wave equation—where renormalized vanishing data converges in law to a non-trivial stochastic limit—violate this continuity condition. Under the enhanced definition, those results become probabilistic ill-posedness theorems. The paper also argues that variance blowup itself, even without a beyond-variance result, corresponds to failure of smoothness of the solution map at the origin, analogous to how deterministic ill-posedness is demonstrated by failure of C^k-smoothness.","feed_headline":"When random-data PDEs that look well-posed are actually ill-posed","feed_subtitle":"A new continuity test reclassifies recent beyond-variance-blowup results for BBM and quadratic NLW as probabilistic ill-posedness, not well-","key_machinery":"The mechanism is a portmanteau-theorem argument: convergence in law to a non-trivial stochastic limit implies that the probability of the solution norm exceeding any small threshold lambda remains bounded away from zero, which directly violates the almost-sure convergence to zero required by condition (iii').","core_discovery":"The central object is the enhanced notion of probabilistic local well-posedness (Definition 1.8), which augments the standard existence-uniqueness-stability conditions with a 'continuity at the origin' requirement: solutions with scaled, frequency-truncated random data delta_N P_N u_0 must converge to zero as N tends to infinity. The authors show that in the BBM equation (for alpha <= 1/4) and the quadratic nonlinear wave equation on T^2 (for beta >= 1/2), the renormalized data delta_{alpha,N} P_N u_0 produces solutions converging in law to non-trivial stochastic PDEs, which by Proposition 2.1 violates the continuity condition and yields probabilistic ill-posedness (Theorems 2.4 and 2.5). A ","pith_inferences":["If the community adopts Definition 1.8 as the standard, the boundary between probabilistic well-posedness and ill-posedness for several dispersive PDEs shifts to the variance-blowup threshold rather than the probabilistic scaling critical threshold, creating a gap between the two heuristics.","The distinction between 'mild' ill-posedness (variance blowup, failure of smoothness) and 'genuine' ill-posedness (beyond variance blowup, failure of continuity) may parallel the deterministic distinction between failure of C^k-smoothness and failure of continuity of the solution map, suggesting a hierarchy of probabilistic ill-posedness.","For equations where variance blowup has not yet been established in the gap between the well-posedness and scaling-critical thresholds (e.g., quadratic NLS for 1/4 < alpha <= 1/2), the framework predicts that either variance blowup will eventually be found there or the probabilistic scaling heuristic will need revision."],"forward_implications":["Results previously interpreted as extending probabilistic well-posedness past the variance-blowup threshold must instead be read as ill-posedness results under the enhanced definition.","For stochastic PDEs with additive forcing, the same phenomenon manifests as failure of stability in the small-noise limit, blocking law-of-large-numbers-type convergence and large deviation results.","Variance blowup alone, without any beyond-variance convergence result, suffices for mild probabilistic ill-posedness via failure of the C^2 bound on the solution map.","The quadratic NLS on T^2 is shown to be mildly probabilistically ill-posed for alpha <= 1/4, and the intermediate range 1/4 < alpha <= 1/2 remains open.","Higher-dimensional extensions of the quadratic NLW ill-posedness result (d >= 3) are natural next targets."],"fun_headline_variants":["Variance blowup reclassified as probabilistic ill-posedness in dispersive PDEs","A continuity-at-zero test exposes hidden ill-posedness in random-data PDEs","New well-posedness criterion turns variance blowup into ill-posedness proofs","BBM and quadratic NLW fail a stronger probabilistic well-posedness test","Probabilistic well-posedness needs continuity at zero — and these PDEs lack it"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The ill-posedness conclusions depend entirely on accepting the 'continuity at the origin' condition (iii') in Definition 1.8 as a necessary part of probabilistic well-posedness. The authors motivate it by analogy to deterministic Hadamard well-posedness, but it is a proposed standard rather than an established one, and the results are relative to this specific definition.","fun_headline_variants_meta":{"raw":{"variants":["Variance blowup reclassified as probabilistic ill-posedness in dispersive PDEs","A continuity-at-zero test exposes hidden ill-posedness in random-data PDEs","New well-posedness criterion turns variance blowup into ill-posedness proofs","BBM and quadratic NLW fail a stronger probabilistic well-posedness test","Probabilistic well-posedness needs continuity at zero — and these PDEs lack it","Beyond variance blowup: reframing random-data PDE results as ill-posedness","Scaling random data to zero reveals probabilistic ill-posedness in two PDEs","Enhanced well-posedness: when stability at zero fails for random PDEs","Continuity at the origin: the missing axiom for probabilistic well-posedness","Two dispersive PDEs fail continuity at zero under randomized initial data"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":975,"prompt_tokens":469,"completion_tokens":506,"prompt_tokens_details":null},"tokens_in":469,"tokens_out":506,"duration_ms":28073,"temperature":1.0,"reasoning_tokens":316,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T04:25:53.349349+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If one can exhibit, for BBM with alpha <= 1/4 or quadratic NLW with beta >= 1/2, that solutions with scaled frequency-truncated data delta_N P_N u_0 do converge to zero in probability (contradicting the convergence-in-law to a non-trivial stochastic limit), then the ill-posedness claim fails.","supporting_citations":[],"review_version":1}