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First pathwise solution theory for stochastic wave equations with white-in-time noise

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Pathwise local well-posedness of stochastic nonlinear wave equations with multiplicative noise is established in optimal regularity ranges by unifying Fourier restriction norm methods with rough path integration.

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load-bearing objection First pathwise well-posedness for SNLW with multiplicative white-in-time noise, via a genuine unification of Fourier restriction norms and rough path integration. the 2 major comments →

arxiv 2607.07618 v1 pith:GMZNLW6H submitted 2026-07-08 math.AP math.PR

Fourier restriction norm method adapted to controlled paths: stochastic wave equations

Andreia Chapouto , Jiawei Li , Tadahiro Oh This is my paper
classification math.AP math.PR
keywords noisepathspathwisestochastictheorywell-posednessadaptedcase
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper resolves a forty-year-old open problem: how to construct solutions to the stochastic nonlinear wave equation with multiplicative white-in-time noise in a pathwise (deterministic, sample-by-sample) sense, rather than only in the Ito (probabilistic average) sense. The obstacle is that the stochastic forcing term is too temporally irregular to define the integral it requires using classical tools. The authors build a single framework that combines three ingredients: (1) the Fourier restriction norm method adapted to U^p/V^p function spaces, which captures both dispersive decay and temporal regularity at the scaling-critical threshold; (2) the sewing lemma and controlled-path theory from rough integration, which defines integrals against irregular signals; and (3) a random tensor estimate for multiple stochastic integrals, which proves the random operator encoding the noise has the temporal Holder regularity needed. In the white-in-time case (Hurst parameter beta = 1/2), the first-order driver has temporal regularity alpha < 1/2, so the classical Young integral fails; the authors construct a second-order driver and impose a controlled-path structure, then split the unknown into a temporally rough but spatially smooth part and a temporally smooth but spatially rough part to close the estimates. The result achieves essentially sharp spatial regularity, matching the deterministic well-posedness threshold.

Core claim

The central object is the interaction representation u_tilde = S(-t)u(t), which removes the linear wave propagator from the stochastic convolution and exposes the temporal regularity of the noise. At this level, the stochastic term becomes a Young integral (for fractional-in-time noise with beta > 1/2) or a rough integral (for white-in-time noise with beta = 1/2) against an operator-valued random driver X whose Holder regularity is established via the random tensor estimate. The key structural innovation for the rough case at high dimension and critical regularity is the decomposition u = v + w, where v is rougher in time but gains spatial regularity from the driver's spatial smoothing, andw

What carries the argument

Sewing lemma; U^p/V^p function spaces (Koch-Tataru); controlled paths (Gubinelli); random tensor estimate for multiple stochastic integrals with respect to fractional Brownian motions; interaction representation; Strichartz estimates; RDE-PDE system decomposition

Load-bearing premise

The noise operator Phi is assumed to be spatially homogeneous (a Fourier multiplier), and the extension to general Hilbert-Schmidt operators, while expected, is not proved.

What would settle it

If one could exhibit a spatially inhomogeneous noise operator for which the random tensor estimates fail to produce the required Holder regularity of the driver, the restriction to Fourier multipliers would be essential rather than merely convenient.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Pathwise well-posedness enables almost-sure convergence of approximation schemes (mollification, piecewise linear) for stochastic wave equations, which Ito theory alone does not provide.
  • The unified framework extends to other stochastic dispersive PDEs with multiplicative noise, including stochastic NLS and stochastic KdV, where temporal roughness induces spatial regularity loss.
  • The one-dimensional result covers almost space-time white noise, which is optimal for one-parameter rough paths; full space-time white noise requires bi-parameter analysis beyond the current framework.
  • The agreement with Ito solutions (Appendix A) bridges the pathwise and probabilistic theories, ensuring consistency with the established random-field solution theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The spatial smoothing of the wave propagator (absent for Schrodinger) is what makes the driver regularity bounds close without spatial loss; extending to stochastic NLS will require a different mechanism to compensate.
  • The restriction to spatially homogeneous noise (Fourier multiplier operators) may be removable for the Young case but could be essential in the rough case where the second-order driver's tensor structure depends on the multiplier factorization.
  • The framework should extend to Hurst parameters beta > 1/3 using higher-order rough paths, at the cost of requiring higher spatial regularity to control the nonlinear term.
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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 paper establishes pathwise local well-posedness of the stochastic nonlinear wave equation (SNLW) with multiplicative noise on the torus T^d, in both the fractional-in-time (Young) and white-in-time (rough) cases. The main results (Theorems 1.4 and 1.5) achieve essentially sharp spatial regularity, matching the deterministic critical Sobolev index s_crit. The approach combines the Fourier restriction norm method adapted to U^p/V^p spaces (Koch-Tataru) with Young/rough integration via the sewing lemma and controlled paths (Gubinelli). A key technical tool is a random tensor estimate (Lemma 2.12) for multiple stochastic integrals with respect to fractional Brownian motions, used to establish almost sure operator-norm bounds on the random drivers (Propositions 4.1, 4.2). The white-in-time result (Theorem 1.5) resolves a long-standing open problem. Appendix A verifies agreement with Ito solutions.

Significance. This is a substantial and important contribution. Theorem 1.5 gives the first pathwise well-posedness result for SNLW with multiplicative white-in-time noise, resolving a forty-year-old open problem. The regularity thresholds are essentially sharp, and in d=1 the result covers almost space-time white noise, which is optimal for one-parameter rough paths. The paper provides complete, detailed proofs of all main components: the random tensor estimate (Lemma 2.12), driver regularity (Propositions 4.1, 4.2), deterministic nonlinear estimates (Section 5), contraction arguments (Sections 6, 7), and Ito agreement (Appendix A). The unified framework combining U^p/V^p spaces with sewing-based rough integration is a methodological contribution likely to influence subsequent work on stochastic dispersive PDEs. The v-w decomposition (Section 7) for handling the endpoint case (d>=6, k=2, s=s_crit>=1) is a notable technical innovation.

major comments (2)
  1. Proposition 4.2(ii), Eq. (4.34): The bound on the second-order driver X requires summing over dyadic blocks under conditions (4.32)-(4.33), verified through a case-by-case analysis (cases (a.i)-(c.ii), pp. 50-51). The convergence in each case reduces to conditions (4.51)-(4.56), which collectively yield (4.32) and (4.33). While the individual cases appear correctly handled, the presentation is dense and the reader must verify six separate frequency interaction regimes. A brief summary table or a more explicit statement of which condition from (4.51)-(4.56) corresponds to which regime would strengthen the verification. This is not an error but a load-bearing argument where clarity directly affects verifiability.
  2. Lemma 2.12 (random tensor estimate): The proof proceeds by induction on TT* powers (Eq. (2.40)-(2.48)), and the inductive step (Eq. (2.45)) relies on the factorization property of the H^beta norm (stated after (2.45): '||f1(t_B1) f2(t_B2)||_{H^beta_{tB}} = prod ||fj||_{H^beta_{tBj}}'). This factorization is correct for the tensor-product structure of H^beta(R^k_+) as defined in (2.23), but it is used implicitly in a context where the time variables have been identified through the contraction (2.43). A one-sentence justification that the factorization survives the relabelling (2.44) would close a gap that a careful reader would flag.
minor comments (5)
  1. p. 50, Eq. (4.45): The expression appears to list (4.45) twice in the text ('putting (4.45), (4.45), (4.46), and (4.47) together'). One reference should likely be to (4.38) or the L^2 bound on eg.
  2. The notation F^(k) in (4.15) uses k as an integer index for sin/cos, which conflicts with k>=2 denoting the nonlinearity power throughout the paper. Consider using a different symbol (e.g., F^(ell)).
  3. Remark 1.22 (p. 8): The spatial homogeneity assumption Phi = Fourier multiplier is stated as a simplification ('We leave the general case to interested readers'). While the random tensor estimate (Lemma 2.12) is stated in general tensor form, the driver regularity proofs (Propositions 4.1, 4.2) use the multiplier structure in (4.14) and (4.41). A brief remark on where the general Hilbert-Schmidt case would require modification would be helpful.
  4. p. 87, proof of Lemma A.6: The notation fM_{s,sigma} (after (A.38)) and tilde-M_{s,sigma} (after (A.42)) are introduced without explicit definition of the tilde variant; the reader can infer it but a definition would be cleaner.
  5. The paper is long (93 pages). While the level of detail is largely justified, some material in Section 3 (Young/rough integration review) and Appendix C (sewing lemma proof) is standard. The authors could consider condensing these sections in a final version, though this does not affect correctness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the positive assessment. Both comments are well-taken and concern clarity of presentation in load-bearing technical arguments. We address each below.

read point-by-point responses

  1. Referee: Proposition 4.2(ii), Eq. (4.34): The bound on the second-order driver X requires summing over dyadic blocks under conditions (4.32)-(4.33), verified through a case-by-case analysis (cases (a.i)-(c.ii), pp. 50-51). The convergence in each case reduces to conditions (4.51)-(4.56), which collectively yield (4.32) and (4.33). While the individual cases appear correctly handled, the presentation is dense and the reader must verify six separate frequency interaction regimes. A brief summary table or a more explicit statement of which condition from (4.51)-(4.56) corresponds to which regime would strengthen the verification.

    Authors: We agree that the case-by-case analysis on pp. 50–51 is dense and that a summary table mapping each frequency interaction regime to the corresponding condition would significantly aid verifiability. In the revised manuscript, we will add a table (or an equivalent explicit summary) listing, for each of the six cases (a.i)–(c.ii), the frequency regime (in terms of the relative sizes of N, N1, N2, N3, N23), the condition from (4.51)–(4.56) that governs convergence in that regime, and how each condition contributes to the unified conditions (4.32)–(4.33). This is purely a clarification; no mathematical change is needed. revision: yes

  2. Referee: Lemma 2.12 (random tensor estimate): The proof proceeds by induction on TT* powers (Eq. (2.40)-(2.48)), and the inductive step (Eq. (2.45)) relies on the factorization property of the H^beta norm (stated after (2.45): '||f1(t_B1) f2(t_B2)||_{H^beta_{tB}} = prod ||fj||_{H^beta_{tBj}}'). This factorization is correct for the tensor-product structure of H^beta(R^k_+) as defined in (2.23), but it is used implicitly in a context where the time variables have been identified through the contraction (2.43). A one-sentence justification that the factorization survives the relabelling (2.44) would close a gap that a careful reader would flag.

    Authors: We agree that a brief justification is warranted. The factorization property follows from the definition (2.23) of H^beta(R^k_+) as a tensor-product Hilbert space: the norm factorizes across any partition of the time variables. After the relabelling (2.44), the sets B1 and C (whose union is D) and B2 and C (whose union is A) are disjoint collections of time variables, so the H^beta-norm on the combined variable set t_B = t_{B1} ⊔ t_{B2} still factorizes as a product of the H^beta-norms on t_{B1} ⊔ t_C and t_{B2} ⊔ t_C. The relabelling merely reindexes which variables belong to which factor; it does not alter the tensor-product structure. We will add one sentence to this effect after (2.45) in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity found

full rationale

The paper's derivation chain is self-contained. The central random tensor estimate (Lemma 2.12) is proved in full via an induction on TT* powers using the Wiener chaos estimate (Lemma 2.8, attributed to Nelson's hypercontractivity — an external, standard result). The driver bounds (Propositions 4.1, 4.2) are derived from Lemma 2.12 through explicit case-by-case frequency analysis. The well-posedness results (Theorems 1.4, 1.5) follow from deterministic contraction arguments (Sections 6, 7) using the driver bounds and standard Strichartz/Sobolev estimates as input. The sewing lemma (Lemma 3.4) is proved in Appendix C. The self-citation to [144] for the β=1/2 case of Lemma 2.12 is not load-bearing because the paper provides a complete proof. No parameter is fitted to data and then presented as a prediction. No uniqueness theorem from the authors' prior work is invoked to forbid alternatives. The conditions on σ and s emerge from the frequency interaction analysis, not from circular reasoning. The agreement with Itô solutions (Proposition A.5) is proved by direct computation. The derivation is genuinely self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 3 invented entities

The paper introduces minimal new entities, all of which serve clear structural purposes and are supported by independent estimates. The free parameters (alpha, gamma, epsilon) are not fitted to data but are chosen to satisfy regularity constraints. The spatial homogeneity assumption on Phi is the main simplifying axiom that is not fully justified in the general case.

free parameters (3)
  • alpha (temporal regularity exponent) = 1/2 < alpha < beta (Young case); 1/3 < alpha < 1/2 (rough case)
    Chosen close to beta to maximize temporal regularity. Not fitted to data but selected to satisfy the constraint alpha + gamma > 1 (or min(alpha+gamma, 3alpha) > 1 in the rough case).
  • gamma (remainder temporal regularity) = 1/2 < gamma < 1, close to 1/2
    Selected in the rough case to satisfy alpha + gamma > 1 with alpha close to 1/2. Not fitted but chosen to close the contraction argument.
  • epsilon (spatial regularity gain) = small positive
    Small spatial smoothing parameter used in the rough case to handle the d>=6, k=2 case via the v-w decomposition. Its existence is guaranteed by the improved nonlinear estimates (Proposition 5.11).
axioms (5)
  • domain assumption The noise operator Phi is spatially homogeneous: Phi(e_n) = phi_n e_n with phi_{-n} = phi_n (equation 1.22).
    Stated in Section 1.3 to simplify presentation. The authors claim the general case is a straightforward extension but do not prove it.
  • standard math The deterministic NLW is locally well-posed in H^s for s satisfying condition (1.21) (Proposition 1.2).
    Follows from standard contraction arguments using Sobolev embedding (d=1) and Strichartz estimates (d>=2). The paper provides the nonlinear estimates in Section 5.
  • standard math The sewing lemma (Lemma 3.4) holds for functions of bounded p-variation.
    Adapted from Holder setting. Proof included in Appendix C.
  • domain assumption The random tensor estimate (Lemma 2.12) extends to fractional Brownian motions with Hurst parameter 1/2 <= beta < 1.
    Stated as a 'straightforward modification' of the beta=1/2 case from [144]. Full proof provided in Subsection 2.5.
  • standard math The Strichartz estimates on T^d follow from those on R^d via finite speed of propagation (Lemma 5.5).
    Cited from [166]. Standard in the dispersive PDE literature.
invented entities (3)
  • Interaction representation u-hat = S(-t)u(t) independent evidence
    purpose: Removes the linear wave propagator from the stochastic convolution term, enabling pathwise construction via Young/rough integration.
    Standard in Fourier restriction norm method. The X^{s,b}-norm is expressed in terms of this representation (equation 1.28).
  • Operator-valued driver X-hat (equation 1.32) and second-order driver X-double-hat (equation 1.37) independent evidence
    purpose: Two-parameter stochastic processes that serve as the Young/rough drivers for the pathwise construction of the stochastic convolution.
    Their regularity is established in Propositions 4.1 and 4.2 using the random tensor estimate, providing falsifiable bounds on operator norms.
  • v-w decomposition (equation 1.42) independent evidence
    purpose: Splits the unknown into a spatially-smoother temporally-rougher part v and a temporally-smoother spatially-rougher part w, to handle the d>=6, k=2 case.
    The decomposition is justified by the spatial smoothing of the rough driver (Proposition 4.2) and the improved nonlinear estimates (Proposition 5.11).

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Cite this review

Pith. "Pith review of Fourier restriction norm method adapted to controlled paths: stochastic wave equations." pith.science (2026). https://pith.science/paper/GMZNLW6H

@misc{pith2026260707618,
  author       = {Pith},
  title        = {Pith review of: Fourier restriction norm method adapted to controlled paths: stochastic wave equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GMZNLW6H}},
  note         = {Machine review of arXiv:2607.07618}
}
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abstract

We investigate the pathwise well-posedness issue of the stochastic nonlinear wave equation (SNLW) with a multiplicative noise. While the Ito solution theory (= random field solution theory) was established in the '80s, its pathwise well-posedness has remained a challenging open problem for over forty years. By building a unified framework for the Fourier restriction norm method adapted to the $U^p$- and $V^p$-spaces, due to Koch and Tataru (2007), and the Young/rough integration theory via the sewing lemma and controlled paths due to Gubinelli (2004) along with the random tensor estimate for multiple stochastic integrals with respect to (fractional) Brownian motions, we establish pathwise local well-posedness of SNLW in optimal regularity ranges. In particular, in the one-dimensional case with a white-in-time noise, our result covers the case of an almost space-time white noise, which is optimal within the framework of one-parameter rough paths.

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Forward citations

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Reference graph

Works this paper leans on

172 extracted references · 172 canonical work pages · cited by 3 Pith papers · 6 internal anchors

  1. [1]

    Arthur, G

    T. Arthur, G. Li, O. Pocovnicu, T. Zhao,Global well-posedness of the energy-critical defocusing stochastic nonlinear wave equations with multiplicative noises, in preparation

    work page

  2. [2]

    Babin, A.A

    A.V. Babin, A.A. Ilyin, E.S. Titi,On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Appl. Math. 64 (2011), no. 5, 591–648

    work page 2011

  3. [3]

    Balan,The stochastic wave equation with multiplicative fractional noise: a Malliavin calculus approach, Potential Anal

    R.M. Balan,The stochastic wave equation with multiplicative fractional noise: a Malliavin calculus approach, Potential Anal. 36 (2012), no. 1, 1–34

    work page 2012

  4. [4]

    Balan, M

    R.M. Balan, M. Jolis, L. Quer-Sardanyons,SPDEs with affine multiplicative fractional noise in space with index 1 4 < H < 1 2, Electron. J. Probab. 20 (2015), no. 54, 36 pp

    work page 2015

  5. [5]

    Barbu, M

    V. Barbu, M. R¨ ockner,The finite speed of propagation for solutions to nonlinear stochastic wave equations driven by multiplicative noise,J. Differential Equations 255 (2013), no. 3, 560–571

    work page 2013

  6. [6]

    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

  7. [7]

    B´ enyi, T

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

    work page 2019

  8. [8]

    B´ enyi, T

    ´A. B´ enyi, T. Oh, T. Zhao,Fractional Leibniz rule on the torus, Proc. Amer. Math. Soc. 153 (2025), no. 1, 207–221

    work page 2025

  9. [9]

    Biagini, Y

    F. Biagini, Y. Hu, B. Øksendal, T. Zhang,Stochastic calculus for fractional Brownian motion and applications, Probability and its Applications (New York). Springer-Verlag London, Ltd., London, 2008. xii+329 pp. 94 A. CHAPOUTO, J. LI, AND T. OH

    work page 2008

  10. [10]

    Bourgain,Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations

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

    work page 1993

  11. [11]

    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

    work page 1994

  12. [12]

    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

    work page 1996

  13. [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

    work page 2021

  14. [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

    work page 2023

  15. [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

    work page 2024

  16. [16]

    Bringmann, Y

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

    work page 2024

  17. [17]

    Bringmann, J

    B. Bringmann, J. L¨ uhrmann, G. Staffilani,The wave maps equation and Brownian paths, Comm. Math. Phys. 405 (2024), no. 3, Paper No. 60, 115 pp

    work page 2024

  18. [18]

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

    work page 2024

  19. [19]

    Brze´ zniak,On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep

    Z. Brze´ zniak,On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep. 61 (1997), no. 3-4, 245–295

    work page 1997

  20. [20]

    Brze´ zniak, M

    Z. Brze´ zniak, M. Ondrej´ at,Strong solutions to stochastic wave equations with values in Riemannian manifolds, J. Funct. Anal. 253 (2007), no. 2, 449–481

    work page 2007

  21. [21]

    Brze´ zniak, M

    Z. Brze´ zniak, M. Ondrej´ at,Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces, Ann. Probab. 41 (2013), no. 3B, 1938–1977

    work page 2013

  22. [22]

    Brze´ zniak, S

    Z. Brze´ zniak, S. Peszat,Space-time continuous solutions to SPDE’s driven by a homogeneous Wiener process, Studia Math. 137 (1999), no. 3, 261–299

    work page 1999

  23. [23]

    Brze´ zniak, N

    Z. Brze´ zniak, N. Rana,Local solution to an energy critical 2-D stochastic wave equation with exponential nonlinearity in a bounded domain, J. Differential Equations 340 (2022), 386–462

    work page 2022

  24. [24]

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

    work page 2008

  25. [25]

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

    work page 2014

  26. [26]

    Cairoli,Sur une ´ equation diff´ erentielle stochastique, C

    R. Cairoli,Sur une ´ equation diff´ erentielle stochastique, C. R. Acad. Sci. Paris S´ er. A-B 274 (1972), A1739–A1742

    work page 1972

  27. [27]

    Cairoli, J

    R. Cairoli, J. Walsh,Stochastic integrals in the plane, Acta Math. 134 (1975), 111–183

    work page 1975

  28. [28]

    Cairoli, J

    R. Cairoli, J. Walsh,Martingale representations and holomorphic processes, Ann. Probability 5 (1977), no. 4, 511–521

    work page 1977

  29. [29]

    Carmona, D

    R. Carmona, D. Nualart,Random nonlinear wave equations: propagation of singularities, Ann. Probab. 16 (1988), no. 2, 730–751

    work page 1988

  30. [30]

    Carmona, D

    R. Carmona, D. Nualart,Random nonlinear wave equations: smoothness of the solutions, Probab. Theory Related Fields 79 (1988), no. 4, 469–508

    work page 1988

  31. [31]

    Caruana, P

    M. Caruana, P. K. Friz,Partial differential equations driven by rough paths, J. Differential Equations 247 (2009), no. 1, 140–173

    work page 2009

  32. [32]

    Caruana, P

    M. Caruana, P. K. Friz, H. Oberhauser,A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 28 (2011), no. 1, 27–46

    work page 2011

  33. [33]

    Catellier, K

    R. Catellier, K. Chouk,Paracontrolled distributions and the 3-dimensional stochastic quantization equation, Ann. Probab. 46 (2018), no. 5, 2621–2679

    work page 2018

  34. [34]

    Chapouto,A refined well-posedness result for the modified KdV equation in the Fourier-Lebesgue spaces

    A. Chapouto,A refined well-posedness result for the modified KdV equation in the Fourier-Lebesgue spaces. J. Dynam. Differential Equations, 35 (2023), no. 3, 2537–2578

    work page 2023

  35. [35]

    Chapouto, M

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

    work page

  36. [36]

    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

    work page

  37. [37]

    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. FOURIER RESTRICTION NORM METHOD ADAPTED TO CONTROLLED PATHS 95

    work page

  38. [38]

    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

    work page

  39. [39]

    X. Chen, A. Deya, J. Song, S. Tindel,Hyperbolic Anderson model 2: Strichartz estimates and Stratonovich setting, Int. Math. Res. Not. IMRN 2023, no. 21, 18575–18628

    work page 2023

  40. [40]

    X. Chen, A. Deya, J. Song, S. Tindel,Solving the hyperbolic Anderson model 1: Skorohod setting, Ann. Inst. Henri Poincar´ e Probab. Stat. 61 (2025), no. 3, 1794–1814

    work page 2025

  41. [41]

    Cheung, G

    K. Cheung, G. Li, T. Oh,Almost conservation laws for stochastic nonlinear Schr¨ odinger equations, J. Evol. Equ. 21 (2021), 1865–1894

    work page 2021

  42. [42]

    Chouk, M

    K. Chouk, M. Gubinelli,Nonlinear PDEs with modulated dispersion I: Nonlinear Schr¨ odinger equations, Comm. Partial Differential Equations 40 (2015), no. 11, 2047–2081

    work page 2015

  43. [43]

    Rough sheets

    K. Chouk, M. Gubinelli,Rough sheets, arXiv:1406.7748 [math.PR]

    work page internal anchor Pith review Pith/arXiv arXiv

  44. [44]

    Nonlinear PDEs with modulated dispersion II: Korteweg-de Vries equation

    C. Chouk, M. Gubinelli, G. Li, J. Li, T. Oh,Nonlinear PDEs with modulated dispersion II: Korteweg-de Vries equation, arXiv:1406.7675 [math.AP]

    work page internal anchor Pith review Pith/arXiv arXiv

  45. [45]

    Ill-posedness for nonlinear Schrodinger and wave equations

    M. Christ, J. Colliander, T. Tao,Ill-posedness for nonlinear Schr¨ odinger and wave equations, arXiv:math/0311048 [math.AP]

    work page internal anchor Pith review Pith/arXiv arXiv

  46. [46]

    Colliander, T

    J. Colliander, T. Oh,Almost sure well-posedness of the cubic nonlinear Schr¨ odinger equation below L2(T), Duke Math. J. 161 (2012), no. 3, 367–414

    work page 2012

  47. [47]

    Da Prato, A

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

    work page 2003

  48. [48]

    Da Prato, J

    G. Da Prato, J. Zabczyk,Stochastic equations in infinite dimensions, Second edition. Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, Cambridge, 2014. xviii+493 pp

    work page 2014

  49. [49]

    Dalang,Extending the martingale measure stochastic integral with applications to spatially homoge- neous s.p.d.e.’s, Electron

    R.C. Dalang,Extending the martingale measure stochastic integral with applications to spatially homoge- neous s.p.d.e.’s, Electron. J. Probab. 4 (1999), no. 6, 29 pp; Corrections, Electron. J. Probab. 6 (2001), no. 6, 5 pp

    work page 1999

  50. [50]

    Dalang,The stochastic wave equation

    R.C. Dalang,The stochastic wave equation. A minicourse on stochastic partial differential equations, 39–71, Lecture Notes in Math., 1962, Springer-Verlag, Berlin, 2009

    work page 1962

  51. [51]

    Dalang, N.E

    R.C. Dalang, N.E. Frangos,The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (1998), no. 1, 187–212

    work page 1998

  52. [52]

    Dalang, M

    R.C. Dalang, M. Sanz-Sol´ e,H¨ older-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc. 199 (2009), no. 931, vi+70 pp

    work page 2009

  53. [53]

    Dalang, M

    R.C. Dalang, M. Sanz-Sol´ e,Stochastic partial differential equations, space-time white noise and random fields, Springer Monographs in Mathematics. Springer, Cham, 2026. xix+634 pp

    work page 2026

  54. [54]

    de Bouard, A

    A. de Bouard, A. Debussche,A stochastic nonlinear Schr¨ odinger equation with multiplicative noise, Comm. Math. Phys. 205 (1999), no. 1, 161–181

    work page 1999

  55. [55]

    de Bouard, A

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

    work page 2003

  56. [56]

    Delgado-Vences, D

    F.J. Delgado-Vences, D. Nualart, G. Zheng,A central limit theorem for the stochastic wave equation with fractional noise, Ann. Inst. Henri Poincar´ e Probab. Stat. 56 (2020), no. 4, 3020–3042

    work page 2020

  57. [57]

    Y. Deng, A. Nahmod, H. Yue,Optimal local well-posedness for the periodic derivative nonlinear Schr¨ odinger equation, Comm. Math. Phys. 384 (2021), no. 2, 1061–1107

    work page 2021

  58. [58]

    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

    work page 2024

  59. [59]

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

    work page 2022

  60. [60]

    Deya,A nonlinear wave equation with fractional perturbation, Ann

    A. Deya,A nonlinear wave equation with fractional perturbation, Ann. Probab. 47 (2019), no. 3, 1775–1810

    work page 2019

  61. [61]

    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

    work page 2020

  62. [62]

    Deya,On ill-posedness of nonlinear stochastic wave equations driven by rough noise, Stochastic Process

    A. Deya,On ill-posedness of nonlinear stochastic wave equations driven by rough noise, Stochastic Process. Appl. 150 (2022), 215–249

    work page 2022

  63. [63]

    A. Deya, M. Gubinelli, M. Hofmanov´ a, S. Tindel,A priori estimates for rough PDEs with application to rough conservation laws, J. Funct. Anal. 276 (2019), no. 12, 3577–3645

    work page 2019

  64. [64]

    A. Deya, M. Gubinelli, S. Tindel,Non-linear rough heat equations, Probab. Theory Related Fields 153 (2012), no. 1-2, 97–147

    work page 2012

  65. [65]

    Farr´ e, D

    M. Farr´ e, D. Nualart,Nonlinear stochastic integral equations in the plane, Stochastic Process. Appl. 46 (1993), no. 2, 219–239. 96 A. CHAPOUTO, J. LI, AND T. OH

    work page 1993

  66. [66]

    Feyel, A

    D. Feyel, A. de La Pradelle,Curvilinear integrals along enriched paths, Electron. J. Probab. 11 (2006), no. 34, 860–892

    work page 2006

  67. [67]

    Forlano, M

    J. Forlano, M. Okamoto,A remark on norm inflation for nonlinear wave equations, Dyn. Partial Differ. Equ. 17 (2020), no. 4, 3610–381

    work page 2020

  68. [68]

    Forlano, L

    J. Forlano, L. Tolomeo,Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schr¨ odinger equations, J. Eur. Math. Soc. (2025). DOI 10.4171/JEMS/1643

    work page doi:10.4171/jems/1643 2025

  69. [69]

    P.K. Friz, M. Hairer,A course on rough paths, With an introduction to regularity structures.Second edition. Universitext, Springer, Cham, [2020]©2020. xvi+346 pp

    work page 2020

  70. [70]

    P.K. Friz, B. Seeger,Besov rough path analysis, With an appendix by Pavel Zorin-Kranich. J. Differential Equations 339 (2022), 152–231

    work page 2022

  71. [71]

    P.K. Friz, N. B. Victoir,Multidimensional stochastic processes as rough paths, Theory and applications. Cambridge Stud. Adv. Math., 120. Cambridge University Press, Cambridge, 2010. xiv+656 pp

    work page 2010

  72. [72]

    Gaines, T

    J. Gaines, T. J. Lyons,Variable step size control in the numerical solution of stochastic differential equations, SIAM J. Appl. Math. 57 (1997), no. 5, 1455–1484

    work page 1997

  73. [73]

    Ginibre, Y

    J. Ginibre, Y. Tsutsumi, G. Velo,On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151 (1997), no. 2, 384–436

    work page 1997

  74. [74]

    Ginibre, G

    J. Ginibre, G. Velo,Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50–68

    work page 1995

  75. [75]

    Greco, M

    D. Greco, M. Gubinelli, S. Liu, T. Oh,Refined global well-posedness for the periodic modulated Korteweg-de Vries equation, preprint

    work page

  76. [76]

    Gubinelli,Controlling rough paths, J

    M. Gubinelli,Controlling rough paths, J. Funct. Anal. 216 (2004), no. 1, 86–140

    work page 2004

  77. [77]

    Gubinelli,Rough solutions for the periodic Korteweg–de Vries equation, Commun

    M. Gubinelli,Rough solutions for the periodic Korteweg–de Vries equation, Commun. Pure Appl. Anal. 11 (2012), no. 2, 709–733

    work page 2012

  78. [78]

    Gubinelli, P

    M. Gubinelli, P. Imkeller, N. Perkowski,Paracontrolled distributions and singular PDEs, Forum Math. Pi 3 (2015), e6, 75 pp

    work page 2015

  79. [79]

    Gubinelli, H

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

    work page 2018

  80. [80]

    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

    work page 2024

Showing first 80 references.

This paper was first reviewed by glm-5.2 on July 9, 2026.