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Weak solutions of Navier-Stokes Equation with purely discontinuous L\'evy Noise

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Pith's one-line read This paper proves the existence of global weak martingale solutions to the stochastic Navier-Stokes equations in $\mathbb{R}^d$ driven by a compensated Poisson random measure, for $d=2,3$, under Lipschitz and growth conditions on the…

desk verdict The genuine new result is the Hilbert-space representation theorem; the NSE application is honest but conditional on a strong Nemytski-continuity assumption (F.4) that needs case-by-case checking. read the letter →

arxiv 2505.24043 v1 pith:N6JKHYBL submitted 2025-05-29 math.PR math.AP

Zdzis{l}aw Brzeźniak , Tomasz Kosmala , Elżbieta Motyl , Paul Razafimandimby This is my paper
classification math.PRmath.AP MSC 60H1535Q3060G5760G4476D05
keywords Navier-StokesequationsLévynoisePoissonrandommeasuremartingalesolutionpurelydiscontinuousrepresentationtheoremGalerkinapproximationtightness
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 establishes that the incompressible Navier-Stokes equations in the whole space, driven by a purely discontinuous Lévy noise described by a Poisson random measure, admit a global weak martingale solution. The construction goes through Galerkin approximations, a priori estimates, and a Skorokhod-type selection on a nonmetric path space; the new ingredient is that the limiting process is shown to be a purely discontinuous martingale, whose representation as a stochastic integral with respect to a Poisson random measure is then proved directly. A key byproduct is a representation theorem for Hilbert-space valued purely discontinuous martingales, which allows the solution to carry an explicit Poisson random measure with the prescribed Lévy measure. A sympathetic reader should take away that the method provides a different route than previous jump-noise treatments, and that the proof is designed to work for other stochastic PDEs driven by Lévy noise.

What carries the argument

The load-bearing object is the candidate martingale $M(t)=u(t)-u(0)-\int_0^t f(s)\,ds+\int_0^t Au(s)\,ds+\int_0^t B(u(s))\,ds$ on the new probability space, together with its real projections $M^{\phi}=\langle M,\phi\rangle_{U',U}$. The proof shows that each $M^{\phi}$ is purely discontinuous by computing its predictable quadratic variation, then uses a representation theorem for Hilbert-space valued purely discontinuous martingales to write $M$ as an integral against a Poisson random measure with intensity $\mathrm{Leb}\otimes\nu$. Auxiliary smooth functions $a_k$ approximating indicators of annuli in $U'$ supply the compensator identification needed to pass from sums of jumps to the $\nu$-integral in the representation step.

What would settle it

One concrete check is to take a coefficient $F$ satisfying the Lipschitz and growth bounds F.2 and F.5 but failing Assumption F.4, for example a map that is continuous into $H$ but not into $U'$ on the Skorokhod space $D([0,T];U')$, and then test whether the compensated jump measure of the limiting process still has compensator $\int_0^t\nu(\{y:F(s,u(s),y)\in A\})\,ds$ for every Borel set $A$ separated from zero; a set where this identity fails would disprove Proposition 6.20 and hence the main theorem.

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Extended reading notes

Core claim

The central claim is Theorem 3.11: under assumptions A.1, P.1, and F.1-F.5, there exists a martingale solution of the abstract stochastic Navier-Stokes system (3.1). The proof passes from Galerkin solutions $u_n$ to a limit $u$ on a new probability space, then shows that the processes $M$ and $N^{\phi}$, defined from $u$ and the test functions, are square-integrable martingales with respect to the filtration generated by $u$. The decisive step is proving that the real-valued projections $M^{\phi}=\langle M,\phi\rangle_{U',U}$ are purely discontinuous martingales by identifying their predictable quadratic variation as $\int_0^t\int_Y\langle F(s,u(s),y),\phi\rangle_H^2\,\nu(dy)\,ds$ and comparing it with the sum of squared jumps. A martingale representation theorem for Hilbert-space valued purely discontinuous martingales, proved in Appendix D, then produces a Poisson random measure $\eta$ on an extension of the probability space such that $M(t)=\int_0^t\int_Y F(s,u(s-),y)\,\tilde{\eta}(ds,dy)$, which completes the construction of a weak martingale solution.

Load-bearing premise

The proof depends on Assumptions F.3 and F.4, which require the noise coefficient to define continuous Nemytski maps on the path space $Z_T$ into $L^2([0,T]\times Y;U')$ and, for test functions, into $L^2$; these continuity conditions are used to pass the stochastic integral to the limit and to identify the jumps of the limiting martingale, so if either fails the representation step cannot be completed.

Editorial extensions

If this is right

  • If Theorem 3.11 is correct, the 2D and 3D stochastic Navier-Stokes equations with pure jump Lévy noise admit global weak martingale solutions under the stated assumptions on the initial data, forcing, and noise coefficient.
  • The constructed solution includes an explicitly identified Poisson random measure with the prescribed Lévy measure, so the solution's jump noise is not an abstract artifact but a concrete component of the probability space.
  • The Hilbert-space martingale representation theorem proved in Appendix D becomes an available tool for proving existence of martingale solutions of other SPDEs driven by Lévy noise, following the same pattern: pass to a limit, prove pure discontinuity of the limit martingale, then represent it.
  • The proof offers an alternative to earlier jump-noise arguments based on compactness-and-tightness routes, and the authors state it does not rely on a representation step they no longer consider reliable.
  • For linear multiplicative noise of the form $F(t,u,y)=G(t,y)u$, the assumptions reduce to verifiable integrability and continuity conditions on $G$, giving a concrete class of coefficients covered by the theorem.

Reading between the lines

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

  • If the Hilbert-space martingale representation theorem is robust, the same two-stage strategy should transfer to other dissipative stochastic partial differential equations with pure jump noise, such as stochastic reaction-diffusion or hydrodynamic-type systems, provided the analogue of tightness on the relevant path space holds.
  • The continuity Assumptions F.3 and F.4 on the Nemytski maps are likely the most restrictive part for applications; one testable extension would be to give sufficient conditions on $F$ or on the Lévy measure $\nu$ under which these maps are automatically continuous, so that only the natural Lipschitz and growth bounds need verification.
  • A natural next step suggested by the method is to investigate uniqueness of invariant measures for 2D Navier-Stokes driven by finite sums of independent Lévy processes; the explicit Poisson random measure representation supplies a canonical noise structure that may facilitate ergodic analysis.
  • The authors' stated uncertainty about a prior representation step implies that a direct, self-contained verification of the new Appendix D representation theorem would be a useful independent check of the method's foundations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proves the existence of martingale solutions to the stochastic Navier-Stokes equations on R^d (d=2,3) driven by a compensated Poisson random measure with intensity Leb⊗ν. The proof follows a Galerkin-tightness-Skorokhod route: uniform estimates from Motyl [47] give tightness of the laws of the Galerkin approximations on the path space Z_T; after passing to a new probability space, the authors identify the limiting process u and construct two families of martingales M and N^φ. They then prove that M^φ is purely discontinuous and compute its predictable quadratic variation, and use a new representation theorem for Hilbert-space valued purely discontinuous martingales (Appendix D, based on Kallianpur-Xiong) to obtain a Poisson random measure η with intensity ν such that M is the stochastic integral of F(·,u(·−),·) against η̃. Substitution into the definition of M yields the desired weak solution. Appendix D states and proves a substantial new result: a martingale representation theorem for purely discontinuous martingales with values in a separable Hilbert space.

Significance. If correct, the theorem provides an alternative existence proof for Lévy-driven SNSEs that does not rely on the reaction-diffusion method of [10]; it also introduces a martingale representation theorem for purely discontinuous Hilbert-space valued martingales that is of independent interest and potentially reusable for other SPDEs with jump noise. The proof is detailed and largely self-contained, and the paper is explicit about the role of the Nemytski-continuity assumptions F.3/F.4, giving worked examples and a sufficient condition (Lemma 3.7). The main limitation is that the main theorem is exactly as strong as those continuity assumptions, which are not implied by the Lipschitz/growth conditions F.2/F.5; this narrows the class of noise coefficients to which the result applies as stated.

major comments (3)
  1. [Section 6, first paragraph and proof of Theorem 3.11] The statement 'without loss of generality, we can and will assume that the external force f in equation (3.1) is equal to 0' is not justified, and Theorem 3.11 is stated with f ∈ L^{4+γ}([0,T]; V'). Because the equation is nonlinear in u, one cannot absorb f by a simple shift of u. The subsequent proof and the final substitution of (6.73) into (5.27) are performed with f=0, so as written the proof establishes the theorem only for f=0. The fix is local: the finite variation term containing f cancels in the purely discontinuous martingale arguments of Sections 6.2–6.4, so the proof can be carried out with f retained, or the authors should add an explicit argument showing how the general case reduces to f=0. As it stands, this is a load-bearing gap in the proof of the stated theorem.
  2. [Section 3.1 (Assumptions F.3, F.4) and Lemma 6.22] The main theorem is exactly as strong as the Nemytski-continuity assumptions F.3 and F.4. Assumption F.4 is used at the critical step (6.46) in Lemma 6.22 to pass to the limit in the noise term and thereby to identify the compensator of the jump measure of M in Section 6.5; Assumption F.3 is used in Lemma 5.14(ii). These conditions are not implied by the pointwise Lipschitz and growth conditions F.2 and F.5, as the paper itself notes in Remark 3.3. The paper would be significantly strengthened by (i) stating more prominently that F.3/F.4 are additional structural hypotheses, (ii) giving a systematic discussion of classes of coefficients that satisfy F.4, beyond Example 3.8 and the sufficient condition in Lemma 3.7, and (iii) commenting on whether F.4 can fail for natural Lévy noise coefficients that satisfy F.2/F.5. This is a limitation of applicability rather than an internal inconsistency, but it deserves to be addressed.
  3. [Section 6.5, verification of Theorem D.6 assumptions] The verification that the integer-valued random measure η_M associated with the jumps of M is of class (QL) is incomplete. Proposition 6.20 establishes the martingale property for sets A ∈ A0, but Definition 2.3 of (QL) also requires σ-finiteness, i.e., the existence of a countable exhaustion of U'\{0} by sets with finite expectation. This does follow from square integrability of M (for instance, the sets {x : |x| ≥ 1/n} have finite expected counts because E[∑_{s≤T} |ΔM(s)|^2] < ∞), but the argument is not given in the text. The authors should spell out this exhaustion argument.
minor comments (5)
  1. [Section 4.2, Lemma 4.3] The tightness of the laws {L(u_n)} on Z_T is imported from [47, Lemma 5] without proof. Since this is a central step of the construction and [47] is not universally available, a sketch of the argument (e.g., the roles of the compact embedding H → U' in (2.34) and of the estimates (4.2)–(4.3)) would improve self-containedness.
  2. [Equation (5.2)] In the definition of N_{n,φ}, the expression 'PnA unu(s)' appears to be a typo and should read 'PnA un(s)'.
  3. [Example 3.8] The assumption on the Lévy measure is stated only for the integral over R\(-1,1), but the verification of F.4 uses ∫_R y^2 ν(dy) < ∞. Since for a Lévy measure ∫_{|y|<1} y^2 ν(dy) < ∞ automatically, the full second moment condition is indeed implied; the text should clarify this to avoid confusion.
  4. [Lemma 3.4] The proof states that convergence in D([0,T]; U') implies u_n(t) → u(t) for Leb-a.a. t. More precisely, this holds for every t outside the countable set of discontinuities of the limit u; the formulation 'for Leb-a.a. t' is correct but the stronger pointwise description would be clearer.
  5. [Introduction] There are a few typographical errors, e.g., 'exietnce' in the first paragraph of the introduction and 'L\'evy' in the abstract. These should be corrected in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.11 is an existence result under explicit hypotheses, the new representation theorem is proved from external results, and the self-citations to Motyl [47,48] concern prior technical lemmas, not the target claim.

full rationale

The paper's central claim, Theorem 3.11, asserts existence of a martingale solution under Assumptions A.1, P.1, and F.1-F.5. No parameter is fitted to data, and no quantity is renamed as a prediction after being built into the hypotheses. The tightness of the Galerkin laws, Lemma 4.3, is imported from Motyl [47, Lemma 5] without proof, and the a priori estimates in Lemma 4.2 are likewise stated as results from [47]; these are self-citations to the third author's prior work, but they concern separate tightness and estimate statements for the Galerkin approximations, not the martingale-solution existence theorem that is the target of the paper. The proof of the key compensator identification, Proposition 6.20, uses Assumptions F.3 and F.4 as explicit continuity hypotheses on the Nemytski maps ΔF and ΔF_φ; the theorem is conditional on those assumptions, so dependence on F.4 is a matter of hypothesis strength, not circularity. The paper's genuinely new ingredient, the representation theorem D.6 for Hilbert-space-valued purely discontinuous martingales, is proved in Appendix D from external results in Ikeda-Watanabe [34, Theorem II.7.4] and Kallianpur-Xiong [39, Theorem 3.4.6], neither of which assumes the Navier-Stokes existence theorem. The authors' explicit statement that they are no longer convinced of part (iii) of Theorem C.1 in their previous paper [10] further shows that the present proof does not lean on the prior method. Thus the derivation chain reduces to standard tightness arguments plus an externally sourced representation theorem, and no load-bearing step exhibits a definitional identity, fitted-input-as-prediction, or self-citation chain that makes the conclusion equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim does not introduce fitted constants or invented entities. It relies on classical stochastic calculus and on the structural assumptions P.1, F.1-F.5, and A.1. The most fragile inputs are the continuity assumptions F.3-F.4 and the imported tightness result from Motyl [47].

assumptions (7)
  • domain assumption Assumption P.1: (Y,Y) is a standard measurable space and ν is σ-finite.
    Used to guarantee the point-process and Poisson random measure representation machinery applies; the authors note in Remark 3.1 that this may be relaxable to Blackwell spaces.
  • domain assumption Assumption F.2: F maps H × Y to H with ν-Lipschitz and linear growth in H.
    Provides well-posedness of the Galerkin SDEs and the uniform square integrability estimates used throughout the proof.
  • domain assumption Assumption F.5: F satisfies an 8+2γ moment growth condition in H for some γ > 0.
    Provides the higher-order a priori estimates in Lemma 4.2 and the uniform integrability needed in the martingale limit passages.
  • domain assumption Assumptions F.3 and F.4: the Nemytski maps associated with F are continuous on Z_T into L^2([0,T]×Y; H or U').
    These are essential for passing the noise term to the limit and for identifying the compensator of the limiting martingale; they are not automatic consequences of F.2.
  • domain assumption Assumption A.1: u0 ∈ H and f ∈ L^{4+γ}(0,T; V'), with the same γ as in F.5.
    Required for the energy estimates in Lemma 4.2 and for the integrability of the force term in the Galerkin equations.
  • standard math There exists a Hilbert space U compactly embedded into V_s, with H embedded compactly into U'.
    This construction is cited from [17] and is used in the tightness argument, specifically Lemma 4.3 and the embedding (2.34).
  • standard math Classical stochastic calculus tools: Itô formula for semimartingales, Burkholder-Davis-Gundy inequalities, Skorokhod-Jakubowski theorem, and the Kallianpur-Xiong representation theorem.
    Invoked from [34], [35], [38], [39], and [18] to derive Galerkin estimates, compactness, and the final martingale representation.

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Pith. "Pith review of Weak solutions of Navier-Stokes Equation with purely discontinuous L\'evy Noise." pith.science (2026). https://pith.science/paper/N6JKHYBL

@misc{pith2026250524043,
  author       = {Pith},
  title        = {Pith review of: Weak solutions of Navier-Stokes Equation with purely discontinuous L\'evy Noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N6JKHYBL}},
  note         = {Machine review of arXiv:2505.24043}
}
read the original abstract

In this paper we prove the existence of weak martingale solutions to the stochastic Navier-Stokes Equations driven by pure jump L\'evy processes. Our proof consists of two parts. In the first one, mostly classical, we recall a priori estimates, from the paper by the third named author, for solutions to suitable constructed Galerkin approximations and we use the Jakubowski-Skorokhod Theorem to find a sequence of processes on a new probability space convergent point-wise to a limit process. In the second one, we show that the limit process is a weak martingale solution to the SNSEs by using an approach of Kallianpur and Xiong. The core of this method consists of a proof that a certain natural process on the new probability space is a purely discontinuous martingale and then to use a suitable representation theorem. In this way we propose a method of proving solutions to stochastic PDEs which is different from the method used recently by the first and fourth named author in their joint paper with E.\ Hausenblas, see \cite{Brz+Haus+Raza_2018_reaction_diffusion}.

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Works this paper leans on

67 extracted references · 66 canonical work pages

  1. [47]

    E. Motyl. Stochastic Navier-Stokes equations driven by Lévy noise in un- bounded 3D domains.Potential Anal., 38(3):863–912, 2013

    work page 2013

  2. [48]

    E. Motyl. Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains – abstract framework and applications. Stochastic Process. Appl., 124(6):2052–2097, 2014

    work page 2014

  3. [10]

    Brzeźniak, E

    Z. Brzeźniak, E. Hausenblas, and P. A. Razafimandimby. Stochastic reaction- diffusion equations driven by jump processes.Potential Anal., 49(1):131–201, 2018

    work page 2018

  4. [1]

    R. A. Adams. Sobolev spaces, volume 65 ofPure and Applied Mathematics. Academic Press, New York-San Francisco-London, second edition, 1975

    work page 1975

  5. [2]

    Agresti and M

    A. Agresti and M. Veraar. Stochastic Navier-Stokes equations for turbulent flows in critical spaces.Comm. Math. Phys., 405(2):Paper No. 43, 57, 2024

    work page 2024

  6. [3]

    Bensoussan and R

    A. Bensoussan and R. Temam. Équations stochastiques du type Navier- Stokes. J. Functional Analysis, 13:195–222, 1973

    work page 1973

  7. [4]

    Convergence of probability measures

    P.Billingsley. Convergence of probability measures. WileySeriesinProbability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1999

    work page 1999

  8. [5]

    Billingsley.Probability and measure

    P. Billingsley.Probability and measure. Wiley Series in Probability and Statis- tics. John Wiley & Sons, Inc., Hoboken, NJ, 2012

    work page 2012

Show all 67 references
  1. [6]

    V. I. Bogachev.Measure theory. vol. I. Springer-Verlag, Berlin, 2007

    2007

  2. [7]

    V. I. Bogachev.Measure theory. vol. II. Springer-Verlag, Berlin, 2007

    2007

  3. [8]

    Brzeźniak, M

    Z. Brzeźniak, M. Capiński, and F. Flandoli. Stochastic Navier-Stokes equa- tions with multiplicative noise.Stochastic Anal. Appl., 10(5):523–532, 1992

    1992

  4. [9]

    Brzeźniak, B

    Z. Brzeźniak, B. Ferrario, and M. Zanella. Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equation. Nonlinearity, 37(1):Paper No. 015001, 66, 2024. 117

    2024

  5. [11]

    Brzeźniak, E

    Z. Brzeźniak, E. Hausenblas, and P. A. Razafimandimby. Stochastic Reaction- diffusion Equations Driven by Jump Processes.Potential Anal., 49(1):131– 201, 2018

    2018

  6. [13]

    Brzeźniak, E

    Z. Brzeźniak, E. Hausenblas, and J. Zhu. 2D stochastic Navier-Stokes equa- tions driven by jump noise.Nonlinear Anal., 79:122–139, 2013

    2013

  7. [14]

    Brzeźniak, F

    Z. Brzeźniak, F. Hornung, and U. Manna. Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise.Stoch. Partial Differ. Equ., Anal. Comput., 8(1):1–53, 2020

    2020

  8. [15]

    Brzeźniak, W

    Z. Brzeźniak, W. Liu, and J. Zhu. Strong solutions for SPDE with locally monotonecoefficientsdrivenbyLévynoise. Nonlinear Anal. Real World Appl., 17:283–310, 2014

    2014

  9. [16]

    Brzeźniak and E

    Z. Brzeźniak and E. Motyl. Existence of solutions of the stochastic boussinesq equations. Global Stoch. Anal., 1(2):175–216, 2011

    2011

  10. [17]

    Existenceofamartingalesolutionofthestochastic Navier-Stokes equations in unbounded 2D and 3D domains.J

    Z.BrzeźniakandE.Motyl. Existenceofamartingalesolutionofthestochastic Navier-Stokes equations in unbounded 2D and 3D domains.J. Differential Equations, 254(4):1627–1685, 2013

    2013

  11. [18]

    Brzeźniak and M

    Z. Brzeźniak and M. Ondreját. Stochastic geometric wave equations with val- ues in compact Riemannian homogeneous spaces.Ann. Probab., 41(3B):1938– 1977, 2013

    1938

  12. [19]

    Well-posednessandlargedeviationsfor2D stochastic Navier-Stokes equations with jumps.J

    Z.Brzeźniak, X.Peng, andJ.Zhai. Well-posednessandlargedeviationsfor2D stochastic Navier-Stokes equations with jumps.J. Eur. Math. Soc. (JEMS), 25(8):3093–3176, 2023

    2023

  13. [20]

    Brzeźniak and S

    Z. Brzeźniak and S. Peszat. Strong local and global solutions for stochastic Navier-Stokes equations. InInfinite dimensional stochastic analysis (Amster- dam, 1999), volume 52 ofVerh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., pages 85–98. R. Neth. Acad. Arts Sci., Amsterdam, 2000

    1999

  14. [21]

    Brzeźniak and S

    Z. Brzeźniak and S. Peszat. Stochastic two dimensional Euler equations.Ann. Probab., 29(4):1796–1832, 2001. 118

    2001

  15. [22]

    Brzeźniak and N

    Z. Brzeźniak and N. Rana. Local solution to an energy critical 2-D stochas- tic wave equation with exponential nonlinearity in a bounded domain. J. Differential Equations, 340:386–462, 2022

    2022

  16. [23]

    R. M. Chen, D. Wang, and H. Wang. Martingale solutions for the three- dimensional stochastic nonhomogeneous incompressible Navier-Stokes equa- tions driven by Lévy processes.J. Funct. Anal., 276(7):2007–2051, 2019

    2007

  17. [24]

    Crisan, D

    D. Crisan, D. D. Holm, J.-M. Leahy, and T. Nilssen. Variational principles for fluid dynamics on rough paths.Adv. Math., 404(part A):Paper No. 108409, 75, 2022

    2022

  18. [25]

    Z. Dong, W. V. Li, and J. Zhai. Stationary weak solutions for stochastic 3D Navier-Stokes equations with Lévy noise.Stoch. Dyn., 12(1):1150006, 22, 2012

    2012

  19. [26]

    Dong and YinChao Xie

    Z. Dong and YinChao Xie. Global solutions of stochastic 2D Navier-Stokes equations with Lévy noise.Sci. China Ser. A, 52(7):1497–1524, 2009

    2009

  20. [27]

    Z. Dong, L. Xu, and X. Zhang. Invariant measures of stochastic 2D Navier- Stokes equations driven byα-stable processes. Electron. Commun. Probab., 16:678–688, 2011

    2011

  21. [28]

    Dong and J

    Z. Dong and J. Zhai. Martingale solutions and Markov selection of stochas- tic 3D Navier-Stokes equations with jump. J. Differential Equations , 250(6):2737–2778, 2011

    2011

  22. [29]

    El Karoui and J.-P

    N. El Karoui and J.-P. Lepeltier. Représentation des processus ponctuels multivariés à l’aide d’un processus de Poisson.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 39(2):111–133, 1977

    1977

  23. [30]

    S. N. Ethier and T. G. Kurtz.M. ov processes Characterization and conver- gence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986

    1986

  24. [31]

    Flandoli and D

    F. Flandoli and D. G¸ atarek. Martingale and stationary solutions for stochas- tic Navier-Stokes equations. Probab. Theory Related Fields, 102(3):367–391, 1995

    1995

  25. [32]

    Flandoli and D

    F. Flandoli and D. Luo. High mode transport noise improves vorticity blow- up control in 3D Navier-Stokes equations. Probab. Theory Related Fields, 180(1-2):309–363, 2021. 119

    2021

  26. [33]

    Grigelionis

    B. Grigelionis. The representation of integer-valued random measures as stochastic integrals over the Poisson measure.Litovsk. Mat. Sb., 11:93–108, 1971

    1971

  27. [34]

    Ikeda and S

    N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes, volume 24 ofNorth-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, second edition, 1989

    1989

  28. [35]

    Jacod and A

    J. Jacod and A. N. Shiryaev.Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, second edition, 2003

    2003

  29. [36]

    Jakubowski

    A. Jakubowski. On the Skorokhod topology.Ann. Inst. H. Poincaré Probab. Statist., 22(3):263–285, 1986

    1986

  30. [37]

    Jakubowski

    A. Jakubowski. The almost sure Skorokhod representation for subsequences in nonmetric spaces.Teor. Veroyatnost. i Primenen. reprinted in Theory Probab. Appl. 42 (1997), no. 1, 167–174 (1998), 42(1):209–216, 1997

    1997

  31. [38]

    Kallenberg.Foundations of modern probability

    O. Kallenberg.Foundations of modern probability. Probability and its Appli- cations. Springer-Verlag, New York, second edition, 2002

    2002

  32. [39]

    Kallianpur and J

    G. Kallianpur and J. Xiong. Stochastic differential equations in infinite- dimensional spaces, volume 26 ofInstitute of Mathematical Statistics Lecture Notes—Monograph Series. Institute of Mathematical Statistics, Hayward, CA, 1995

    1995

  33. [40]

    Kumar and M

    A. Kumar and M. T. Mohan. Well-posedness of a class of stochastic par- tial differential equations with fully monotone coefficients perturbed by Lévy noise. Anal. Math. Phys., 14(3):Paper No. 44, 59, 2024

    2024

  34. [41]

    Lindvall

    T. Lindvall. Weak convergence of probability measures and random functions in the function spaceD(0, ∞). J. Appl. Probability, 10:109–121, 1973

    1973

  35. [42]

    Lions and E

    J.-L. Lions and E. Magenes.Non-homogeneous boundary value problems and applications. Vol. I. Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth

    1972

  36. [43]

    Marinelli and M

    C. Marinelli and M. Röckner. On maximal inequalities for purely discontin- uous martingales in infinite dimensions. InSéminaire de Probabilités XLVI, volume 2123 ofLecture Notes in Math., pages 293–315. Springer, Cham, 2014

    2014

  37. [44]

    W. McLean. Strongly elliptic systems and boundary integral equations. Cam- bridge University Press, Cambridge, 2000. 120

    2000

  38. [45]

    Métivier

    M. Métivier. Semimartingales: A course on stochastic processes, volume 2 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin-New York, 1982

    1982

  39. [46]

    Mikulevicius and B

    R. Mikulevicius and B. L. Rozovskii. GlobalL2-solutions of stochastic Navier- Stokes equations. Ann. Probab., 33(1):137–176, 2005

    2005

  40. [49]

    Ndongmo Ngana, G

    A. Ndongmo Ngana, G. Deugoué, and T. Tachim Medjo. Weak solution of a stochastic 3D nonlocal Cahn-Hilliard-Navier-Stokes systems with shear- dependent viscosity.Stochastics, 95(4):521–580, 2023

    2023

  41. [50]

    Nguyen, K

    P. Nguyen, K. Tawri, and R. Temam. Nonlinear stochastic parabolic par- tial differential equations with a monotone operator of the Ladyzenskaya- Smagorinsky type, driven by a Lévy noise.J. Funct. Anal., 281(8):Paper No. 109157, 74, 2021

    2021

  42. [51]

    Ondreját

    M. Ondreját. personal communication

  43. [52]

    K. R. Parthasarathy. Probability measures on metric spaces. Probability and Mathematical Statistics, No. 3. Academic Press, Inc., New York-London, 1967

    1967

  44. [53]

    Well-posednessofstochastic2Dhydrodynamics type systems with multiplicative Lévy noises.Electron

    X.Peng, J.Yang, andJ.Zhai. Well-posednessofstochastic2Dhydrodynamics type systems with multiplicative Lévy noises.Electron. J. Probab., 27:Paper No. 55, 31, 2022

    2022

  45. [54]

    Revuz and M

    D. Revuz and M. Yor.Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, corrected printing of third edition, 2005

    2005

  46. [55]

    W. Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991

    1991

  47. [56]

    Sakthivel and S

    K. Sakthivel and S. S. Sritharan. Martingale solutions for stochastic Navier- Stokes equations driven by Lévy noise.Evol. Equ. Control Theory, 1(2):355– 392, 2012. 121

    2012

  48. [57]

    A. V. Skorohod. Limit theorems for stochastic processes.Teor. Veroyatnost. i Primenen., 1:289–319, 1956

    1956

  49. [58]

    H. Tanaka. On the uniqueness of Markov process associated with the Boltz- mann equation of Maxwellian molecules. InProceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), pages 409–425. Wiley, New York-...

    1976

  50. [59]

    R. Temam. Navier-Stokes equations. AMS Chelsea Publishing, Providence, RI, 2001. Theory and numerical analysis, Reprint of the 1984 edition

    2001

  51. [60]

    A. W. van der Vaart.Asymptotic statistics, volume 3 ofCambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 1998

    1998

  52. [61]

    M. C. Veraar. Stochastic integration in Banach spaces and applications to parabolic evolution equations. Delft University of Technology, 2006. Ph.D. Thesis–TU Delft

    2006

  53. [62]

    M. J. Vishik and A. V. Fursikov. Mathematical problems of statistical hy- dromechanics, volume 9 ofMathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the 1980 Russian original [ MR0591678] by D. A. Leites

    1988

  54. [63]

    Xu and T

    T. Xu and T. Zhang. Large deviation principles for 2-D stochastic Navier- Stokes equations driven by Lévy processes.J. Funct. Anal., 257(5):1519–1545, 2009

    2009

  55. [64]

    Yaroslavtsev.Martingales And Stochastic Calculus In Banach Spaces

    I. Yaroslavtsev.Martingales And Stochastic Calculus In Banach Spaces. Delft University of Technology, 2019. Ph.D. Thesis–TU Delft

    2019

  56. [65]

    Yaroslavtsev

    I. Yaroslavtsev. Burkholder-Davis-Gundy inequalities in UMD Banach spaces. Comm. Math. Phys., 379(2):417–459, 2020

    2020

  57. [66]

    I. S. Yaroslavtsev. On strongly orthogonal martingales in UMD Banach spaces. Probab. Math. Statist., 41(1):153–171, 2021

    2021

  58. [67]

    Zhu.A study of SPDES w.r.t

    J. Zhu.A study of SPDES w.r.t. compensated Poisson random measures and related topics. University of York, 2010. Ph.D. Thesis

    2010

  59. [68]

    J. Zhu, Z. Brzeźniak, and Wei Liu. Maximal inequalities and exponential estimates for stochastic convolutions driven by Lévy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations.SIAM J. Math. Anal., 51(3):2121–2167, 2019. 122

    2019

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