{"id":"b0cc0db9-1124-4e02-a52e-192c442db6dd","arxiv_id":"2505.24043","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Weak martingale solutions exist for the 2D and 3D stochastic Navier-Stokes equations driven by pure jump Lévy noise, under Lipschitz, growth, and continuity assumptions on the noise coefficient.","lead":"This paper proves that the fluid equations known as Navier-Stokes, when pushed by random jumps rather than smooth noise, have weak solutions in two and three dimensions under natural conditions on the jumps. It matters because it supplies a general proof route, based on representing jump martingales, that can be reused for other stochastic partial differential equations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 3.11 hinges on the Nemytski-continuity Assumptions F.3/F.4 in the limit passage of Lemma 6.22, a condition not implied by F.2/F.5; the theorem's validity is conditional on verifying these assumptions.","rationale":"The paper gives a detailed, mostly coherent proof of a substantial conditional result. The Galerkin approximations, a priori estimates, tightness, Skorokhod–Jakubowski transfer, and the martingale properties of M and N^φ are either proved in the text or imported from Motyl [47]. The reader's conditional verdict is appropriate. I focused on the limit passage that deploys the paper's novelty, the representation of the limiting jump martingale. That passage, specifically Lemma 6.22 and Proposition 6.20, uses the Nemytski-continuity assumptions F.3 and F.4 in an essential way. These are not consequences of the Lipschitz and growth conditions F.2 and F.5, and the paper does not offer a reduction. This is not an internal contradiction, but it is the least secure condition: a strong, self-standing hypothesis that must be verified for each noise coefficient. The imported tightness lemma is also fragile, but it is a standard lemma in the same framework as [47]; F.4 is more directly tied to the new martingale-representation argument. The concrete test of attempting to prove (6.39) under F.2/F.5 alone would settle whether the theorem can be broadened, but as it stands the argument is conditional on F.4. Therefore the reader's verdict of CONDITIONAL, with moderate confidence, should stand unchanged.","tokens_in":78474,"tokens_out":28117,"duration_ms":266905,"concrete_test":"Test the essential convergence (6.39) for a coefficient F(t,u,y)=g(y)Φ(u), where g∈L^2(Y,ν) and Φ:H→H is H-Lipschitz but not continuous from U' to U' (e.g., Φ(u)=φ(⟨u,e_1⟩_H)e_1 with a Lipschitz φ that is discontinuous at some point in the weak topology). Choose a sequence z_n→z in Z_T that converges pointwise in U' but not in H, and compute the left-hand side of (6.39). If the limit does not vanish, F.4 is genuinely load-bearing and cannot be replaced by F.2/F.5; if it vanishes, one may attempt to reprove Lemma 6.22 under F.2/F.5 alone and thereby weaken the assumptions of Theorem 3.11.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central obstruction is the dependence of the representation-theoretic step on Assumptions F.3 and F.4. In Lemma 6.22, the convergence (6.39) is proved by bounding the difference and using F.4 to send the term in (6.46) to zero: continuity of F as a map from Z_T into L^2([0,T]×Y;U'). This Nemytski continuity is not a consequence of the H-norm Lipschitz and growth conditions F.2 and F.5; those only control pointwise H-values, while F.4 requires joint continuity in the weak/D-topology of Z_T. If F.4 fails for a concrete noise coefficient, the identification of the compensator of the jump measure of M with the image of ν under F(·,u(·−),·) breaks down, and the martingale representation theorem D.6 cannot be invoked. Thus the main theorem is exactly as strong as its F.3/F.4 hypotheses, and every application must verify them independently. The imported tightness lemma (Lemma 4.3 from [47]) is a secondary fragile point, but F.4 sits at the heart of the paper's new method.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":78719,"tokens_out":14588,"duration_ms":136488,"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":[{"comment":"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.","section":"Section 6, first paragraph and proof of Theorem 3.11"},{"comment":"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.","section":"Section 3.1 (Assumptions F.3, F.4) and Lemma 6.22"},{"comment":"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.","section":"Section 6.5, verification of Theorem D.6 assumptions"}],"minor_comments":[{"comment":"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.","section":"Section 4.2, Lemma 4.3"},{"comment":"In the definition of N_{n,φ}, the expression 'PnA unu(s)' appears to be a typo and should read 'PnA un(s)'.","section":"Equation (5.2)"},{"comment":"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.","section":"Example 3.8"},{"comment":"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.","section":"Lemma 3.4"},{"comment":"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.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper's core idea is sound and the representation theorem in Appendix D is a genuine contribution. The main issue that blocks acceptance as written is the unjustified 'without loss of generality f=0' reduction, which leaves the stated theorem (with f ∈ L^{4+γ}) unproved for nonzero f. This is fixable by keeping the f terms throughout, but it is a load-bearing point that must be addressed. The reliance on the Nemytski-continuity assumptions F.3/F.4 and the imported tightness lemma are limitations that should also be clarified. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: the paper's real contribution is not the existence theorem for NSEs with Levy noise, which largely overlaps with Motyl's earlier work, but the Hilbert-space representation theorem for purely discontinuous martingales (Thm D.6). That theorem is proved at length, and I don't see a gap in it. If you work on Levy-driven SPDEs, this is the part to read.\n\nThe paper does what it claims and is honest about it. It says up front that Section 4 follows [47]/[48], and that the new part is the martingale representation plus the limit passage. It also explicitly flags doubt about part (iii) of Theorem C.1 in [10], which explains the new method and is refreshing candor.\n\nSoft spots, in proportion. The reader's concern about F.3/F.4 is legitimate: Lemma 6.22, which identifies the compensator of the jump measure, uses F.4 directly in the limit argument, and F.4 is not implied by the Lipschitz/growth conditions F.2/F.5. So the main theorem is exactly as strong as the assumption that F defines a continuous Nemytski map on Z_T into L^2([0,T]xY; U'). The authors give one multiplicative example where it works, but the abstract theorem is conditional on this strong condition; a referee should ask for a cleaner sufficient condition or an explicit statement that applications must verify it separately.\n\nSecond, the tightness lemma 4.3 is imported from [47] without proof. That is reliance on prior work, not circularity, but it means the paper is not self-contained at a key step. A referee should check that [47, Lemma 5] really covers the present Galerkin setting.\n\nNo fitted parameters, no circular reasoning. The conditional verdict is fair. For the right reader, the paper deserves serious referee time. I'd send it to a strong math.PR journal with instructions to verify F.4 and the imported tightness lemma.","headline":"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.","tokens_in":79264,"tokens_out":2595,"would_cite":true,"duration_ms":26701,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","35Q30","60G57","60G44","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Navier-Stokes equations","Lévy noise","Poisson random measure","martingale solution","purely discontinuous martingale","martingale representation theorem","Galerkin approximation","tightness"],"falsifier":"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.","tokens_in":78289,"feed_emoji":"🌊","tokens_out":5597,"duration_ms":58073,"temperature":0.7,"pith_summary":"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.","feed_headline":"Weak martingale solutions exist for jump-noise Navier-Stokes","feed_subtitle":"A representation theorem builds the Poisson random measure explicitly, opening a new proof route for SPDEs with Lévy noise.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Galerkin approximations, the a priori estimates, and the tightness criterion used to obtain the convergent subsequence.","marker":"[47]"},{"why":"Provides the unbounded-domain variational framework, the auxiliary Hilbert spaces $U,V,H$, and convergence lemmas for the nonlinear term used in the limit passage.","marker":"[17]"},{"why":"Supplies the representation theorem for purely discontinuous martingales that the paper adapts to Hilbert-space valued processes, along with compensator identification lemmas.","marker":"[39]"},{"why":"Provides the theory of Poisson random measures, stochastic integration with respect to compensated random measures, and the Itô formula applied to the Galerkin equations.","marker":"[34]"},{"why":"Gives the Skorokhod representation theorem for nonmetric spaces, used to select the convergent subsequence on the path space $Z_T$.","marker":"[37]"},{"why":"States the topological version of the Skorokhod theorem applied through Corollary 4.5.","marker":"[18]"},{"why":"Supplies definitions and properties of quadratic variation and pure discontinuity for Hilbert-space valued martingales used in Section 6.","marker":"[45]"}],"fun_headline_variants":["Weak martingale solutions proven for NS with Lévy jumps","Jump Lévy noise yields weak martingale solutions","Existence of weak solutions to NS with pure jump Lévy","New proof route for NS martingale solutions with jumps","Pure jump Lévy noise: weak martingale solutions exist"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Weak martingale solutions proven for NS with Lévy jumps","Jump Lévy noise yields weak martingale solutions","Existence of weak solutions to NS with pure jump Lévy","New proof route for NS martingale solutions with jumps","Pure jump Lévy noise: weak martingale solutions exist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000171,"raw_usage":{"total_tokens":1302,"prompt_tokens":1009,"completion_tokens":293,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":214}},"tokens_in":625,"tokens_out":293,"duration_ms":3235,"temperature":1.0,"reasoning_tokens":214,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:37:58.068469+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Galerkin approximations, the a priori estimates, and the tightness criterion used to obtain the convergent subsequence."},{"cited_title":"Existenceofamartingalesolutionofthestochastic Navier-Stokes equations in unbounded 2D and 3D domains.J","cited_arxiv_id":null,"evidence_quote":"Provides the unbounded-domain variational framework, the auxiliary Hilbert spaces $U,V,H$, and convergence lemmas for the nonlinear term used in the limit passage."},{"cited_title":"Kallianpur and J","cited_arxiv_id":null,"evidence_quote":"Supplies the representation theorem for purely discontinuous martingales that the paper adapts to Hilbert-space valued processes, along with compensator identification lemmas."},{"cited_title":"Ikeda and S","cited_arxiv_id":null,"evidence_quote":"Provides the theory of Poisson random measures, stochastic integration with respect to compensated random measures, and the Itô formula applied to the Galerkin equations."},{"cited_title":"Jakubowski","cited_arxiv_id":null,"evidence_quote":"Gives the Skorokhod representation theorem for nonmetric spaces, used to select the convergent subsequence on the path space $Z_T$."},{"cited_title":"Brzeźniak and M","cited_arxiv_id":null,"evidence_quote":"States the topological version of the Skorokhod theorem applied through Corollary 4.5."},{"cited_title":"Métivier","cited_arxiv_id":null,"evidence_quote":"Supplies definitions and properties of quadratic variation and pure discontinuity for Hilbert-space valued martingales used in Section 6."}],"review_version":1}