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REVIEW 2 major objections 5 minor 95 references

On probabilistic ill-posedness

T0 review · 2 major / 5 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Random data PDEs that look well-posed are actually ill-posed

desk verdict Conceptual note reinterpreting variance blowup results as probabilistic ill-posedness; sound logic, worth a referee read the letter →

arxiv 2607.07628 v1 pith:CCXUMKAZ submitted 2026-07-08 math.AP math.PR

Tadahiro Oh , Nikolay Tzvetkov This is my paper · ORCID
classification math.APmath.PR MSC 35R6060H1560H3035Q3535L7135Q53
keywords probabilisticwell-posednessill-posednessvarianceblowupdispersivePDErandominitialdataBBMequationnonlinearwavecontinuityattheorigin
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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').

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. 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
  2. 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.
minor comments (5)
  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: 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,

    Authors: 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: yes

  2. Referee: 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.

    Authors: 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: yes

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity found; self-citation of [59, 58] is load-bearing but non-circular

full rationale

The paper's logical chain is: (1) introduce Definition 1.8 with continuity-at-origin condition (iii'); (2) prove Proposition 2.1, a standard measure-theoretic bridge (Egoroff + portmanteau) showing that if lim sup P(||u_N|| > λ) > 0 then (iii') is violated; (3) cite convergence-in-law results from [59, 58] (co-authored by the present authors) to verify that condition via (2.9) and (2.18); (4) conclude ill-posedness (Theorems 2.4, 2.5). Each link is independently meaningful. Definition 1.8 is new and not defined in terms of the cited results. Proposition 2.1 is a genuine deduction, not a tautology: a.s. convergence to 0 implies convergence in probability and hence in law, so convergence in law to a non-trivial limit contradicts (iii') by uniqueness of weak limits. The cited works [59, 58] provide convergence-in-law theorems (u_N → u ≠ 0) that were proved independently of Definition 1.8 — they do not assume or use the present paper's framework. The self-citation is therefore real evidence (externally falsifiable mathematical results with their own proofs), not a circular input. Section 3's argument (variance blowup → unbounded second Picard iterate → failure of derivative bound (3.4) → mild ill-posedness under Definition 3.2) is also a straightforward logical deduction. The only concern is that the ill-posedness conclusions are relative to a definition the authors propose, but that is a conceptual choice, not circularity. Score 2 reflects the load-bearing self-citation without any reduction-by-construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a conceptual/mathematical note. It introduces no new physical entities or fitted parameters. The axioms are standard domain assumptions from probabilistic PDE theory, plus the paper's own proposed definition which is argued to be natural. The 'beyond variance blowup' results from companion papers are treated as established facts.

assumptions (4)
  • domain assumption The standard notion of probabilistic local well-posedness includes existence, uniqueness, and stability under frequency truncation (properties (i) and (ii) in Subsection 1.2).
    This is the standard framework in the field, established by Bourgain and others, used as the baseline for the enhanced definition.
  • domain assumption For monomial nonlinearities, the enhanced data set satisfies the homogeneity property Xi_k(delta * u_0) = delta^{lambda_k} * Xi_k(u_0) (equation 1.17).
    This structural property is used to derive the convergence Xi(delta * u_0) -> 0 as delta -> 0 (equation 1.18), which motivates the continuity at the origin condition.
  • domain assumption The 'beyond variance blowup' results from [59, 58] establish convergence in law of solutions with renormalized data to non-trivial stochastic PDE limits.
    These results are treated as given inputs (Theorems in companion papers) from which the ill-posedness conclusions are derived.
  • ad hoc to paper The enhanced notion of probabilistic well-posedness (Definition 1.8), particularly the continuity at the origin condition (iii'), is a natural and correct criterion to impose.
    This is the new definition introduced by the paper. Its adoption as a standard is a conceptual choice that the paper argues for by analogy to deterministic theory.

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Cite this review

Pith. "Pith review of On probabilistic ill-posedness." pith.science (2026). https://pith.science/paper/CCXUMKAZ

@misc{pith2026260707628,
  author       = {Pith},
  title        = {Pith review of: On probabilistic ill-posedness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CCXUMKAZ}},
  note         = {Machine review of arXiv:2607.07628}
}
abstract

In this note, we introduce an enhanced notion of probabilistic well-posedness for dispersive PDEs with random initial data by imposing stability at the origin as the amplitude of randomization tends to $0$. We then use this notion to re-interpret recent works on "beyond variance blowup" for dispersive PDEs, by the authors with their collaborators (2025, 2026), as probabilistic ill-posedness results. By drawing an analogy to the failure of $C^k$-smoothness of a solution map in the deterministic setting, we interpret variance blowup results as mild probabilistic ill-posedness.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

95 extracted references · 95 canonical work pages

  1. [58]

    G. Li, J. Li, S. Liu, T. Oh, N. Tzvetkov,Probabilistic well-posedness of dispersive PDEs beyond variance blowup II: quadratic nonlinear wave equation, preprint

    work page

  2. [1]

    Barashkov, P

    N. Barashkov, P. Laarne,Transition state theory for the hyperbolicϕ 4 model, Electron. J. Probab. 31 (2026), Paper No. 14, 58 pp

    work page 2026

  3. [2]

    Bass,Stochastic processes, Cambridge Series in Statistical and Probabilistic Mathematics, 33

    R. Bass,Stochastic processes, Cambridge Series in Statistical and Probabilistic Mathematics, 33. Cam- bridge University Press, Cambridge, 2011. xvi+390 pp

    work page 2011

  4. [3]

    Benjamin, J.L

    T.B. Benjamin, J.L. Bona, J.J. Mahony,Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A 272 (1972), no. 1220, 47–78

    work page 1972

  5. [4]

    B´ enyi, T

    ´A. B´ enyi, T. Oh, O. Pocovnicu,On the probabilistic Cauchy theory of the cubic nonlinear Schr¨ odinger equation onR d,d≥3, Trans. Amer. Math. Soc. Ser. B 2 (2015), 1–50

    work page 2015

  6. [5]

    B´ enyi, T

    ´A. B´ enyi, T. Oh, O. Pocovnicu,Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schr¨ odinger equation onRd, Trans. Amer. Math. Soc. Ser. B 6 (2019), 114–160

    work page 2019

  7. [6]

    B´ enyi, T

    ´A. B´ enyi, T. Oh, O. Pocovnicu,On the probabilistic Cauchy theory for nonlinear dispersive PDEs, Land- scapes of Time-Frequency Analysis. 1–32, Appl. Numer. Harmon. Anal., Birkh¨ auser/Springer, Cham, 2019

    work page 2019

  8. [7]

    J.L. Bona, M. Dai,Norm Inflation for the BBM equation, J. Math. Anal. Appl. 446 (2016), 879–885

    work page 2016

Show all 95 references
  1. [8]

    J. Bona, N. Tzvetkov,Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst. 23 (2009), no. 4, 1241–1252

    2009

  2. [9]

    Bourgain,Fourier transform restriction phenomena for certain lattice subsets and applications to non- linear evolution equations

    J. Bourgain,Fourier transform restriction phenomena for certain lattice subsets and applications to non- linear evolution equations. I. Schr¨ odinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156

    1993

  3. [10]

    Bourgain,Periodic nonlinear Schr¨ odinger equation and invariant measures, Comm

    J. Bourgain,Periodic nonlinear Schr¨ odinger equation and invariant measures, Comm. Math. Phys. 166 (1994), no. 1, 1–26

    1994

  4. [11]

    Bourgain,Invariant measures for the 2D-defocusing nonlinear Schr¨ odinger equation, Comm

    J. Bourgain,Invariant measures for the 2D-defocusing nonlinear Schr¨ odinger equation, Comm. Math. Phys. 176 (1996), no. 2, 421–445

    1996

  5. [12]

    Bourgain,Periodic Korteweg de Vries equation with measures as initial data, Selecta Math

    J. Bourgain,Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997), no. 2, 115–159. ON PROBABILISTIC ILL-POSEDNESS 23

    1997

  6. [13]

    Bringmann,Almost sure local well-posedness for a derivative nonlinear wave equation, Int

    B. Bringmann,Almost sure local well-posedness for a derivative nonlinear wave equation, Int. Math. Res. Not. IMRN 2021, no. 11, 8657–8697

    2021

  7. [14]

    Bringmann,Introduction to deterministic and random dispersive equation, lecture notes from Summer School on PDEs and Randomness (2023), Max Planck Institute Leipzig

    B. Bringmann,Introduction to deterministic and random dispersive equation, lecture notes from Summer School on PDEs and Randomness (2023), Max Planck Institute Leipzig. https://files-www.mis.mpg.de/mpi-typo3/events-files/slides 764.pdf

    2023

  8. [15]

    Bringmann,Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlin- earity II: dynamics, J

    B. Bringmann,Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlin- earity II: dynamics, J. Eur. Math. Soc. (JEMS) 26 (2024), no. 6, 1933–2089

    2024

  9. [16]

    Bringmann, Y

    B. Bringmann, Y. Deng, A. Nahmod, H. Yue,Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation,Invent. Math. 236 (2024), no. 3, 1133–1411

    2024

  10. [17]

    E. Brun, G. Li, R. Liu,Global well-posedness of the energy-critical stochastic nonlinear wave equations, J. Differential Equations 397 (2024), 316–348

    2024

  11. [18]

    Brze´ zniak, B

    Z. Brze´ zniak, B. Go ldys, M. Ondrej´ at, N. Rana,Large deviations for (1+1)-dimensional stochastic geo- metric wave equation, J. Differential Equations 325 (2022), 1–69

    2022

  12. [19]

    N. Burq, N. Camps, C. Sun, N. Tzvetkov,Probabilistic well-posedness for the nonlinear Schr¨ odinger equation on the 2d sphere I: positive regularities, arXiv:2404.18229 [math.AP]

    arXiv

  13. [20]

    N. Burq, N. Camps, C. Sun, N. Tzvetkov,Gauge transforms, random averaging operator ansatz and improved probabilistic well-posedness for the radial NLS on the 3dball, arXiv:2606.07010 [math.AP]

    arXiv

  14. [21]

    N. Burq, N. Tzvetkov,Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 (2008), no. 3, 449–475

    2008

  15. [22]

    N. Burq, N. Tzvetkov,Probabilistic well-posedness for the cubic wave equation,J. Eur. Math. Soc. (JEMS) 16 (2014), no. 1, 1–30

    2014

  16. [23]

    Cerrai, M

    S. Cerrai, M. Xie,On the small noise limit in the Smoluchowski-Kramers approximation of nonlinear wave equations with variable friction, Trans. Amer. Math. Soc. 376 (2023), no. 11, 7651–7689

    2023

  17. [24]

    Chapouto, M

    A. Chapouto, M. Gubinelli, G. Li, J. Li, T. Oh,Nonlinear PDEs with modulated dispersion III: multi- plicative noises, preprint

  18. [25]

    Chapouto, J

    A. Chapouto, J. Li, F.G. Longmou-Moffo, T. Oh, M. Okamoto,Pathwise local well-posedness of the stochastic Korteweg-de Vries equation with a multiplicative Young noise, preprint

  19. [26]

    Chapouto, J

    A. Chapouto, J. Li, T. Oh,Fourier restriction norm method adapted to controlled paths: stochastic wave equations, preprint

  20. [27]

    Chapouto, J

    A. Chapouto, J. Li, T. Oh,Fourier restriction norm method adapted to controlled paths II: stochastic nonlinear Schr¨ odinger equations on the Euclidean space, in progress

  21. [28]

    Chapouto, J

    A. Chapouto, J. Li, T. Oh, G. Zheng,Pathwise well-posedness of the stochastic nonlinear Schr¨ odinger equations with multiplicative noises, in preparation

  22. [29]

    Christ, J

    M. Christ, J. Colliander, T. Tao,Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003), no. 6, 1235–1293

    2003

  23. [30]

    Colliander, T

    J. Colliander, T. Oh,Almost sure well-posedness of the cubic nonlinear Schr¨ odinger equation belowL 2(T), Duke Math. J. 161 (2012), no. 3, 367–414

    2012

  24. [31]

    Da Prato, A

    G. Da Prato, A. Debussche,Strong solutions to the stochastic quantization equations, Ann. Probab. 31 (2003), no. 4, 1900–1916

    2003

  25. [32]

    de Bouard, A

    A. de Bouard, A. Debussche,The stochastic nonlinear Schr¨ odinger equation inH 1,Stochastic Anal. Appl. 21 (2003), no. 1, 97–126

    2003

  26. [33]

    de Bouard, A

    A. de Bouard, A. Debussche, Y. Tsutsumi,Periodic Solutions of the Korteweg-de Vries Equation Driven by White Noise,SIAM J. Math. Anal. 36 (2004), no. 3, 815–855

    2004

  27. [34]

    Y. Deng, A. Nahmod, H. Yue,Invariant Gibbs measures and global strong solutions for nonlinear Schr¨ odinger equations in dimension two, Ann. of Math. 200 (2024), no. 2, 399–486

    2024

  28. [35]

    Y. Deng, A. Nahmod, H. Yue,Random tensors, propagation of randomness, and nonlinear dispersive equations, Invent. Math. 228 (2022), no. 2, 539–686

    2022

  29. [36]

    Y. Deng, A. Nahmod, H. Yue,The probabilistic scaling paradigm, Vietnam J. Math. 52 (2024), no. 4, 1001–1015

    2024

  30. [37]

    Deya,On a non-linear 2D fractional wave equation, Ann

    A. Deya,On a non-linear 2D fractional wave equation, Ann. Inst. Henri Poincar´ e Probab. Stat. 56 (2020), no. 1, 477–501

    2020

  31. [38]

    Forlano,Almost sure global well posedness for the BBM equation with infiniteL 2 initial data, Discrete Contin

    J. Forlano,Almost sure global well posedness for the BBM equation with infiniteL 2 initial data, Discrete Contin. Dyn. Syst. 40 (2020), no. 1, 267–318

    2020

  32. [39]

    Forlano, T

    J. Forlano, T. Oh, Y. Wang,Stochastic cubic nonlinear Schr¨ odinger equation with almost space-time white noise, J. Aust. Math. Soc. 109 (2020), no. 1, 44–67. 24 T. OH AND N. TZVETKOV

    2020

  33. [40]

    Forlano, Y

    J. Forlano, Y. Zine,Invariant Gibbs dynamics for the hyperbolic sinh-Gordon model, arXiv:2602.14996 [math.AP]

  34. [41]

    Grafakos,Classical Fourier analysis

    L. Grafakos,Classical Fourier analysis. Third edition. Graduate Texts in Mathematics, 249. Springer, New York, 2014. xviii+638 pp

    2014

  35. [42]

    Greco, P

    D. Greco, P. Sosoe, T. Oh, Y. Wang,Normal form approach to unconditional well-posedness of the periodic stochastic Korteweg-de Vries equation with an additive noise, preprint

  36. [43]

    Greco, T

    D. Greco, T. Oh, K. Tsugawa,Unconditional well-posedness of the stochastic Korteweg-de Vries equation on the real line, preprint

  37. [44]

    Gubinelli, M

    M. Gubinelli, M. Hairer, T. Oh, Y. Zine,A simple construction of the sine-Gordon model via stochastic quantization, J. Lond. Math. Soc. 112 (2025), no. 1, Paper No. e70214

    2025

  38. [45]

    Gubinelli, H

    M. Gubinelli, H. Koch, T. Oh,Renormalization of the two-dimensional stochastic nonlinear wave equa- tions, Trans. Amer. Math. Soc. 370 (2018), no. 10, 7335–7359

    2018

  39. [46]

    Gubinelli, H

    M. Gubinelli, H. Koch, T. Oh,Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity, J. Eur. Math. Soc. (JEMS) 26 (2024), no. 3, 817–874

    2024

  40. [47]

    Gubinelli, H

    M. Gubinelli, H. Koch, T. Oh, L. Tolomeo,Global dynamics for the two-dimensional stochastic nonlinear wave equations, Int. Math. Res. Not. 2022, no. 21, 16954–16999

    2022

  41. [48]

    Z. Guo, S. Kwon, T. Oh,Poincar´ e-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS, Comm. Math. Phys. 322 (2013), no. 1, 19–48

    2013

  42. [49]

    Hadamard,Sur les probl` emes aux d´ eriv´ ees partielles et leur signification physique, The Princeton University Bulletin (1902), 49–52

    J. Hadamard,Sur les probl` emes aux d´ eriv´ ees partielles et leur signification physique, The Princeton University Bulletin (1902), 49–52

    1902

  43. [50]

    Hadamard,Les probl` emes aux limites dans la th´ eorie des ´ equations aux d´ eriv´ ees partielles, J

    J. Hadamard,Les probl` emes aux limites dans la th´ eorie des ´ equations aux d´ eriv´ ees partielles, J. Phys. Theor. Appl. 6 (1907), 202–241

    1907

  44. [51]

    Hadamard,Lectures on Cauchy’s Problem in Linear Partial Differential Equations.New Haven: Yale University Press; London: Humphrey Milford; Oxford: University Press

    J. Hadamard,Lectures on Cauchy’s Problem in Linear Partial Differential Equations.New Haven: Yale University Press; London: Humphrey Milford; Oxford: University Press. VIII u. 316 S. (1923)

    1923

  45. [52]

    Hairer,Renormalisation in the presence of variance blowup, Ann

    M. Hairer,Renormalisation in the presence of variance blowup, Ann. Probab. 53 (2025), no. 5, 1958–1985

    2025

  46. [53]

    Herr,Well-posedness results for dispersive equations with derivative nonlinearities, Dissertation, Uni- versit¨ at Dortmund, Dortmund, Germany, 2006

    S. Herr,Well-posedness results for dispersive equations with derivative nonlinearities, Dissertation, Uni- versit¨ at Dortmund, Dortmund, Germany, 2006

    2006

  47. [54]

    Hoshino,KPZ equation with fractional derivatives of white noise, Stoch

    M. Hoshino,KPZ equation with fractional derivatives of white noise, Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016), no. 4, 827–890

    2016

  48. [55]

    Kaneshiro,A new proof of the abstract random tensor estimate by Deng, Nahmod, and Yue, arXiv:2512.02250 [math.PR]

    C. Kaneshiro,A new proof of the abstract random tensor estimate by Deng, Nahmod, and Yue, arXiv:2512.02250 [math.PR]

  49. [56]

    Kato,On nonlinear Schr¨ odinger equations

    T. Kato,On nonlinear Schr¨ odinger equations. II.H s-solutions and unconditional well-posedness,J. Anal. Math. 67 (1995), 281–306

    1995

  50. [57]

    Kenig, G

    C.E. Kenig, G. Ponce, L. Vega,A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573–603

    1996

  51. [59]

    G. Li, J. Li, T. Oh, N. Tzvetkov,Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation, arXiv:2509.02344 [math.AP]

    arXiv

  52. [60]

    Liu,On the probabilistic well-posedness of the two-dimensional periodic nonlinear Schr¨ odinger equation with the quadratic nonlinearity|u| 2, J

    R. Liu,On the probabilistic well-posedness of the two-dimensional periodic nonlinear Schr¨ odinger equation with the quadratic nonlinearity|u| 2, J. Math. Pures Appl. 171 (2023), 75–101

    2023

  53. [61]

    R. Liu, T. Oh,Sharp local well-posedness of the two-dimensional periodic nonlinear Schr¨ odinger equation with a quadratic nonlinearity|u| 2, Math. Res. Lett. 31 (2024), no. 1, 255–277

    2024

  54. [62]

    R. Liu, N. Tzvetkov, Y. Wang,Existence, uniqueness, and universality of global dynamics for the fractional hyperbolicΦ 4 3-model, arXiv:2311.00543 [math.AP]

    arXiv

  55. [63]

    Statistical mechanics of nonlinear wave equations. IV. Cubic Schr¨ odinger

    H.P. McKean,Statistical mechanics of nonlinear wave equations. IV. Cubic Schr¨ odinger, Comm. Math. Phys. 168 (1995), no. 3, 479–491.Erratum: “Statistical mechanics of nonlinear wave equations. IV. Cubic Schr¨ odinger”, Comm. Math. Phys. 173 (1995), no. 3, 675

    1995

  56. [64]

    Molinet,Global well-posedness in the energy space for the Benjamin-Ono equation on the circle, Math

    L. Molinet,Global well-posedness in the energy space for the Benjamin-Ono equation on the circle, Math. Ann. 337 (2007), no. 2, 353–383

    2007

  57. [65]

    Oh,Periodic stochastic Korteweg-de Vries equation with additive space-time white noise, Anal

    T. Oh,Periodic stochastic Korteweg-de Vries equation with additive space-time white noise, Anal. PDE 2 (2009), no. 3, 281–304

    2009

  58. [66]

    Oh,White noise for KdV and mKdV on the circle, Harmonic analysis and nonlinear partial differential equations, 99–124, RIMS Kˆ okyˆ uroku Bessatsu, B18, Res

    T. Oh,White noise for KdV and mKdV on the circle, Harmonic analysis and nonlinear partial differential equations, 99–124, RIMS Kˆ okyˆ uroku Bessatsu, B18, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010

    2010

  59. [67]

    Oh,A remark on norm inflation with general initial data for the cubic nonlinear Schr¨ odinger equations in negative Sobolev spaces, Funkcial

    T. Oh,A remark on norm inflation with general initial data for the cubic nonlinear Schr¨ odinger equations in negative Sobolev spaces, Funkcial. Ekvac. 60 (2017), 259–277. ON PROBABILISTIC ILL-POSEDNESS 25

    2017

  60. [68]

    T. Oh, M. Okamoto,Comparing the stochastic nonlinear wave and heat equations: a case study, Electron. J. Probab. 26 (2021), Paper No. 9, 44 pp

    2021

  61. [69]

    T. Oh, M. Okamoto, O. Pocovnicu, N. Tzvetkov,A remark on randomization of a general function of negative regularity, Proc. Amer. Math. Soc. Ser. B 11 (2024), 538–554

    2024

  62. [70]

    T. Oh, M. Okamoto, L. Tolomeo,FocusingΦ 4 3-model with a Hartree-type nonlinearity.Mem. Amer. Math. Soc. 304 (2024), no. 1529

    2024

  63. [71]

    T. Oh, M. Okamoto, L. Tolomeo,Stochastic quantization of theΦ 3 3-model.Mem. Eur. Math. Soc., 16 EMS Press, Berlin, 2025, viii+145 pp

    2025

  64. [72]

    T. Oh, M. Okamoto, N. Tzvetkov,Uniqueness and non-uniqueness of the Gaussian free field evolution under the two-dimensional Wick ordered cubic wave equation, Ann. Inst. Henri Poincar´ e Probab. Stat. 60 (2024), no. 3, 1684–1728

    2024

  65. [73]

    T. Oh, O. Pocovnicu, N. Tzvetkov,Probabilistic local Cauchy theory of the cubic nonlinear wave equation in negative Sobolev spaces, Ann. Inst. Fourier (Grenoble) 72 (2022) no. 2, 771–830

    2022

  66. [74]

    T. Oh, J. Quastel,On Cameron-Martin theorem and almost sure global existence, Proc. Edinb. Math. Soc. 59 (2016), 483–501

    2016

  67. [75]

    T. Oh, J. Quastel, P. Sosoe,Global dynamics for the stochastic KdV equation with white noise as initial data, Trans. Amer. Math. Soc. Ser. B 11 (2024), 420–460

    2024

  68. [76]

    T. Oh, T. Robert, P. Sosoe, Y. Wang,On the two-dimensional hyperbolic stochastic sine-Gordon equation, Stoch. Partial Differ. Equ. Anal. Comput. 9 (2021), 1–32

    2021

  69. [77]

    T. Oh, T. Robert, P. Sosoe, Y. Wang,Invariant Gibbs dynamics for the dynamical sine-Gordon model, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 5, 1450–1466

    2021

  70. [78]

    T. Oh, T. Robert, N. Tzvetkov,Stochastic nonlinear wave dynamics on compact surfaces, Ann. H. Lebesgue 6 (2023), 161–223

    2023

  71. [79]

    T. Oh, T. Robert, Y. Wang,On the parabolic and hyperbolic Liouville equations, Comm. Math. Phys. 387 (2021), no. 3 1281–1351

    2021

  72. [80]

    T. Oh, L. Thomann,Invariant Gibbs measures for the 2-ddefocusing nonlinear wave equations, Ann. Fac. Sci. Toulouse Math. 29 (2020), no. 1, 1–26

    2020

  73. [81]

    T. Oh, N. Tzvetkov, Y. Wang,Solving the 4NLS with white noise initial data, Forum Math. Sigma 8 (2020), e48, 63 pp

    2020

  74. [82]

    T. Oh, Y. Wang,Normal form approach to the one-dimensional periodic cubic nonlinear Schr¨ odinger equation in almost critical Fourier–Lebesgue spaces, J. Anal. Math. 143 (2021), no. 2, 723–762

    2021

  75. [83]

    T. Oh, Y. Wang, Y. Zine,Three-dimensional stochastic cubic nonlinear wave equation with almost space- time white noise, Stoch. Partial Differ. Equ. Anal. Comput. 10 (2022), 898–963

    2022

  76. [84]

    Panthee,On the ill-posedness result for the BBM equation, Discrete Contin

    M. Panthee,On the ill-posedness result for the BBM equation, Discrete Contin. Dyn. Syst. 30 (2011), 253–259

    2011

  77. [85]

    Pocovnicu,Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation onR d,d= 4and5, J

    O. Pocovnicu,Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation onR d,d= 4and5, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 8, 2521–2575

    2017

  78. [86]

    Roum´ egoux,A symplectic non-squeezing theorem for BBM equation, Dyn

    D. Roum´ egoux,A symplectic non-squeezing theorem for BBM equation, Dyn. Partial Differ. Equ. 7 (2010), 289–305

    2010

  79. [87]

    Y. Shao, J. Li, T. Oh,A remark on pathwise well-posedness of the 1-dstochastic heat equation, preprint

  80. [88]

    C. Sun, N. Tzvetkov,New examples of probabilistic well-posedness for nonlinear wave equations, J. Funct. Anal. 278 (2020), no. 2, 108322, 47 pp

    2020

  81. [89]

    C. Sun, N. Tzvetkov,Concerning the pathological set in the context of probabilistic well-posedness, C. R. Math. Acad. Sci. Paris 358 (2020), no. 9-10, 989–999

    2020

  82. [90]

    C. Sun, N. Tzvetkov,Gibbs measure dynamics for the fractional NLS, SIAM J. Math. Anal. 52 (2020), no. 5, 4638–4704

    2020

  83. [91]

    C. Sun, N. Tzvetkov,Refined probabilistic global well-posedness for the weakly dispersive NLS, Nonlinear Anal. 213 (2021), Paper No. 112530, 91 pp

    2021

  84. [92]

    C. Sun, N. Tzvetkov, W. Xu,Weak universality for a class of nonlinear wave equations, to appear in Ann. Inst. Fourier (Grenoble)

  85. [93]

    Tzvetkov,Remark on the local ill-posedness for KdV equation, C

    N. Tzvetkov,Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris S´ er. I Math. 329 (1999), no. 12, 1043–1047

    1999

  86. [94]

    Tzvetkov,Random data wave equations, Singular random dynamics, 221–313, Lecture Notes in Math., 2253, Fond

    N. Tzvetkov,Random data wave equations, Singular random dynamics, 221–313, Lecture Notes in Math., 2253, Fond. CIME/CIME Found. Subser., Springer, Cham, [2019],©2019

    2019

  87. [95]

    Zine,Hyperbolic sine-Gordon model beyond the first threshold, arXiv:2504.07944 [math.AP]

    Y. Zine,Hyperbolic sine-Gordon model beyond the first threshold, arXiv:2504.07944 [math.AP]. 26 T. OH AND N. TZVETKOV Tadahiro Oh, School of Mathematics, The University of Edinburgh, and The Maxwell In- stitute for the Mathematical Sciences, James Clerk Maxwell Building, The K...

    arXiv

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