{"id":"1fce0874-e434-4ede-bb57-a422541dd843","arxiv_id":"2502.00189","paper_version":4,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For SPDEs with cylindrical Wiener noise in Banach spaces, weak existence plus pathwise uniqueness is equivalent to strong existence plus joint weak uniqueness under flexible path-space and integrability assumptions.","lead":"This paper proves a general equivalence for stochastic PDEs in infinite-dimensional spaces: weak existence plus pathwise uniqueness is equivalent to strong existence plus joint weak uniqueness. The result covers several standard notions of solution, including analytically strong, analytically weak, and mild solutions, and includes cases that earlier theorems could not handle.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Polish path-space assumption is the load-bearing restriction; the abstract's 'arbitrary Banach spaces' claim for analytically weak solutions is broader than what Assumption 2.1 delivers.","rationale":"The reader identified Assumption 2.1's Polish continuous-path hypothesis as the weakest point, and the proof confirms that this is where the theory's scope is determined. I agree with that identification. The internal argument from Kurtz's theorem is carefully constructed: the measurable representation of the stochastic integral, the translation of equation conditions into distributional constraints, and the compatibility arguments all appear sound for the class of spaces satisfying Assumption 2.1. I do not see an internal mathematical contradiction in Theorem 3.1. However, the abstract's statement that analytically weak solutions hold in arbitrary Banach spaces goes beyond what the proof establishes. Conditions (3) and (5) alone allow weakly continuous paths that are not strongly continuous in the state space, and Assumption 2.1 excludes exactly those paths unless an auxiliary compact embedding is available. The paper's Example 2.13c is an honest illustration of this extra work, but it relies on compactness of the manifold. The proposed test with a weakly continuous but not norm-continuous path in ℓ^2 would make the limitation concrete. Thus the theorem is acceptable for its stated framework, but the advertised scope should be qualified: the results are conditional on the existence of a Polish path space B embedded in C(¯IT;Z), and for analytically weak solutions this is an additional regularity/compactness assumption unless the solution notion is defined accordingly.","tokens_in":53273,"tokens_out":37816,"duration_ms":371384,"concrete_test":"Take T=1, U=R, Y=Z=ℓ^2, b=σ=0, and let f∈C_w([0,T];ℓ^2) be given by f(t)=e_{⌊1/t⌋} for t>0 and f(0)=0. Check whether f satisfies condition (3) and the weak equation (5) with D=Z∗ for the equation du=0: it does, yet f∉C([0,T];ℓ^2), so no Polish B⊂C([0,T];ℓ^2) satisfying Assumption 2.1 can contain it. Then verify whether any alternative Z (e.g. a Hilbert space under a compact injection) can accommodate the same weak formulation with all z∗∈Z∗; because Z∗ is smaller than ℓ^2, the original weak equation is not recovered. This settles whether the phrase 'for analytically weak solutions, the results hold in arbitrary Banach spaces' requires an additional compactness/path-regularity assumption.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The theorem is a conditional equivalence: Theorem 3.1 is only as general as Assumption 2.1, and the proof uses the Polish/Borel structure of B in every essential step (Lemma 3.3, Kuratowski arguments in Corollaries 3.4/3.9, Blackwell–Dubins in Theorem 1.6, and the measurability of F_mu in Step 5). The most load-bearing clause is B Polish with B↪→C(¯IT;Z). This forces solutions to have strongly continuous paths in some separable Banach space Z and forces the weak-continuity/positivity conditions to be Borel subsets of such a path space. For analytically weak solutions, condition (5) only gives scalar continuity of ⟨u(·),z∗⟩; it does not produce strong path continuity in Z. Therefore the claimed 'arbitrary Banach spaces' scope is not a consequence of the stochastic-integral representation; it is an extra hypothesis on the solution class. Example 2.13c can encode weak continuity only when a compact embedding (Rellich) upgrades weak continuity to strong continuity. Without compactness, e.g. Y=Z=ℓ^2, the weakly continuous path f(t)=e_{⌊1/t⌋}, f(0)=0, satisfies the usual weak equation with zero coefficients but is not in any B⊂C([0,T];ℓ^2). Whether such paths are meant to be covered determines whether the abstract is accurate. This is a scope limitation, not an internal inconsistency of the proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a Yamada–Watanabe–Engelbert equivalence for a wide class of SPDE solution notions in Banach spaces. Under Assumption 2.1, which fixes a Polish path space B continuously embedded into C(I_T;Z), the paper shows that for any admissible collection C of conditions (covering analytically strong, analytically weak, mild, and weakly mild solutions) the following are equivalent: existence of a C-weak solution plus pathwise uniqueness, existence of a C-strong solution plus joint weak uniqueness, and joint weak uniqueness plus the existence of a single measurable solution map F_μ from Z×W to B (Theorem 3.1). The proof reduces the statement to Kurtz's abstract theorem after showing that all conditions in C can be expressed through the joint law of (u,W). A key tool is a measurable representation of the stochastic integral in martingale type 2 and UMD Banach spaces (Theorem 3.7). The paper also derives an analogue of the classical Yamada–Watanabe theorem (Corollary 3.15) and discusses applications to variational, critical-space, and weak/mild solution settings.","tokens_in":53529,"tokens_out":16909,"duration_ms":180224,"significance":"If the results stand, the paper provides a valuable unification: it covers several existing Yamada–Watanabe theorems (Ondreját, Röckner–Schmuland–Zhang, Kunze) and extends the Yamada–Watanabe–Engelbert equivalence to solution classes for which it was not previously available. The measurable representation of stochastic integrals is a useful contribution in its own right and is the technical heart of the paper. The proof is detailed and does not rely on fitted parameters or on assuming the conclusion. The main caveat is that the scope of the 'arbitrary Banach spaces' claim for analytically weak solutions is narrower than the abstract suggests, because the path space B is required to be Polish and to consist of strongly continuous paths.","major_comments":[{"comment":"The abstract's statement that for analytically weak solutions 'the results hold in arbitrary Banach spaces' is broader than what Assumption 2.1 delivers. In Definition 2.5 every C-weak solution must satisfy u ∈ B a.s., where B is Polish and B embeds continuously into C(I_T;Z), and the proof uses this path-space regularity essentially (Lemma 3.3, Kuratowski arguments in Corollaries 3.4 and 3.9, Blackwell–Dubins representations, and the measurable-selection step in Step 5 of Theorem 3.1). For a solution class whose paths are only weakly continuous, condition (5) gives only scalar continuity of t → ⟨u(t), z*⟩ and does not force strong continuity in Z. Example 2.13c encodes weak continuity only together with strong continuity in a larger space, using compactness (Rellich); in a general Banach space such as ℓ², weakly continuous but not strongly continuous paths are not covered by any B ⊂ C([0,T];Z). Thus the theorem is correct as a conditional statement, but the claimed 'arbitrary Banach spaces' scope should be qualified, e.g. 'arbitrary separable Banach spaces with solution paths constrained to a Polish subspace of strongly continuous paths', or a proof of the stronger claim should be supplied.","section":"Abstract; §2, Assumption 2.1 and Definition 2.5; Theorem 3.1"},{"comment":"The map F in Corollary 3.15(ii) is only shown to be separately measurable: for each z ∈ Z0, F(z,·) is B_t(W)^{P∞}/B_t(B)-measurable, but joint measurability of F: Z0 × W → B is not established. The proof sets F(z,w) = F_{δ_z}(z,w) and does not show that z ↦ F_{δ_z}(z,w) is measurable in z for a fixed w. Consequently, F(u0,W) is not proved to be a random variable for a general random initial condition u0, and Corollary 3.15 does not deliver the 'unique strong solution' notion of [24, Def. 1.6 Chap. 4] or [42, Def. 1.9], which require a measurable solution map in both variables. Either joint measurability should be proved, or the statement should explicitly say that F is only a separately measurable selection and the claim that this is the classical Yamada–Watanabe theorem should be weakened accordingly.","section":"Corollary 3.15"}],"minor_comments":[{"comment":"The definition of ar I_T is not stated clearly enough: since ar I_T is used throughout as the domain of continuous paths and contains 0, it should be defined explicitly as [0,T] for T < ∞ and [0,∞) for T = ∞.","section":"Notation, p. 4"},{"comment":"The UMD version of the representation theorem is stated only for T ∈ (0,∞), while Theorem 3.1 allows T = ∞; the proof handles T = ∞ through the local conditions (2) and (9), but a sentence in the statement of Theorem 3.7 or Corollary 3.9 explaining this localisation would prevent confusion.","section":"Theorem 3.7 and Corollary 3.9"},{"comment":"The identification of μ ∈ P(Z0) with its trivial extension in P(Z) is made in the text immediately before the corollary, but it is easy to miss; since F_μ is a priori defined only for μ ∈ P(Z), this identification should be stated as part of the corollary's assumptions.","section":"Corollary 3.15"}],"recommendation":"major_revision","confidential_remarks":"The main Yamada–Watanabe–Engelbert theorem appears sound and the proof is careful; the issues I raise concern scope and the strength of the 'classical Yamada–Watanabe' claim. After the abstract and Corollary 3.15 are qualified, and assuming the authors either prove or explicitly disclaim joint measurability in Corollary 3.15, the paper would be a strong candidate for acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is one of the cleanest Kurtz-based treatments of the Yamada-Watanabe-Engelbert theorem for SPDEs, and the measurable representation theorem for stochastic integrals is a genuine new ingredient. The main theorem is proved by reducing to Kurtz's abstract theorem with a carefully constructed Gamma, and I did not find a circular step or fitted constants. The framework covers analytically strong, analytically weak, mild, and weakly mild solutions in a single setup, and it re-proves the earlier results of Ondrejat, Roeckner-Schmuland-Zhang, and Kunze. The treatment of the UMD versus M-type-2 stochastic integration is careful, and the appendices fill in the measure-theoretic details rather than hand-waving them.\n\nThe main soft spot is the one the stress-test flags: the Polish path-space assumption B -> C(bar I_T; Z) is load-bearing, and the abstract's claim that \"for analytically weak solutions, the results hold in arbitrary Banach spaces\" is broader than what Assumption 2.1 delivers. Condition (5) only gives scalar continuity of the solution. Without strong continuity in Z, or a compact embedding that upgrades weak continuity to strong continuity as in Example 2.13c, a weakly continuous solution need not lie in any B embedded in C(bar I_T; Z). So the theorem applies to analytically weak solution notions that impose strong path continuity or an upgrade, not to all weakly continuous analytically weak solutions in arbitrary Banach spaces. This is a real scope limitation, but it is not an internal inconsistency: the proof is coherent under its stated assumptions.\n\nThe second softness is less about the argument and more about verification: the proof leans on deep external results in Banach-space stochastic integration, and the measurability steps in Lemma 3.6 and Theorem 3.7 are intricate. I did not find a specific gap, but a specialist in vector-valued stochastic integration should check those steps before publication. This is normal for the area and not a reason to reject.\n\nWho gets value from this: stochastic analysts working on SPDE well-posedness, especially those who want a unified way to think about strong versus weak solution notions and about pathwise uniqueness. I would cite this for the measurable representation theorem and the unified framework, and I would bring it to a reading group despite the length. My recommendation: send it to a serious referee, and ask the author to either tone down the arbitrary-Banach-space sentence in the abstract or clarify exactly which weak-solution notions are covered when the path space is not strongly continuous.","headline":"A serious, well-built Kurtz-based Yamada-Watanabe-Engelbert framework for SPDEs, with a genuinely new measurable representation of stochastic integrals, though the abstract overstates the arbitrary-Banach-space scope for analytically weak solutions.","tokens_in":54072,"tokens_out":2474,"would_cite":true,"duration_ms":28916,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60H05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A unified proof shows that for SPDEs in Banach spaces, weak existence plus pathwise uniqueness is equivalent to the existence of a strong solution plus joint weak uniqueness, and that a single measurable map produces solutions from…","keywords":["Yamada–Watanabe theorem","Yamada–Watanabe–Engelbert theorem","SPDEs in Banach spaces","cylindrical Brownian motion","pathwise uniqueness","joint weak uniqueness","martingale type 2","UMD spaces"],"falsifier":"Test the analytically weak case in a Banach space that is neither UMD nor of martingale type 2, such as $L^1$, with a nontrivial cylindrical Wiener noise: the paper asserts the equivalence holds there, and a failure of pathwise uniqueness or of the strong-solution-map conclusion in such an example would refute the claim.","tokens_in":53045,"feed_emoji":"🎲","tokens_out":10956,"duration_ms":90794,"temperature":0.7,"pith_summary":"This paper proves a Yamada–Watanabe–Engelbert theorem for stochastic partial differential equations in Banach spaces: under one set of assumptions, weak existence plus pathwise uniqueness is equivalent to the existence of a strong solution plus joint weak uniqueness, and also to the existence of a single measurable map that turns any initial value and any cylindrical Brownian motion into a solution. The result covers analytically strong, analytically weak, mild, and weakly mild solution notions in a unified way, and it holds in martingale type 2 or UMD Banach spaces, with the analytically weak case working in arbitrary Banach spaces. The value of the paper is that the classical equivalence, previously proved separately for different SPDE frameworks, follows from one abstract theorem once the stochastic integral admits a measurable representation that depends on the law of the integrand and the noise. That representation is proved here for infinite dimensions and is of independent interest.","feed_headline":"For SPDEs, weak existence plus uniqueness yields a solution map","feed_subtitle":"A single theorem covers mild, weak, and analytically strong SPDE solutions in Banach spaces.","key_machinery":"The load-bearing object is a measurable representation $I$ of the stochastic integral: for any stochastically integrable process $f$ and cylindrical Brownian motion $W$, one has $I(f(\\omega), W(\\omega), \\mathrm{Law}(f,W)) = (\\int_0^\\cdot f\\,dW)(\\omega)$ for almost every $\\omega$. Because the representation takes the joint law of integrand and noise as an argument, the condition 'u solves the SPDE' becomes a property of the joint distribution of $u$ and $W$, which is exactly what is needed to apply an abstract Yamada–Watanabe–Engelbert theorem for compatible solutions. The other central piece is the compatibility structure: a family of $\\sigma$-algebras $\\mathcal{B}^t$ encoding the information of paths up to time $t$, which lets adaptedness and independence of noise increments be read off from the law of $(u, u(0), W)$.","core_discovery":"The central claim is Theorem 3.1. Given a Polish path space $B$ embedded continuously in the space of continuous $Z$-valued paths, and given a collection $C$ of solution conditions (chosen from analytically strong, analytically weak, mild, or weakly mild, with optional moment and a.s. constraints), for a fixed initial law $\\mu$ the following are equivalent: (a) a $C$-weak solution exists and pathwise uniqueness holds; (b) a $C$-strong solution exists and joint weak uniqueness holds; (c) joint weak uniqueness holds and there is a Borel measurable map $F_\\mu: Z\\times W\\to B$ such that, on any filtered probability space, any $U$-cylindrical Brownian motion $W$, and any initial value with law $\\mu$, the pair $(F_\\mu(u_0,W), W)$ is a $C$-strong solution, with paths up to time $t$ depending measurably on the paths of $(u_0,W)$ up to time $t$. The paper also derives the classical Yamada–Watanabe theorem as a corollary, giving a single map $F$ independent of the initial law under an all-initial-laws hypothesis. The proof runs through an abstract Yamada–Watanabe–Engelbert theorem for compatible solutions; the main input that makes the SPDE case work is a measurable representation of the stochastic integral in martingale type 2 or UMD spaces.","pith_inferences":["The law-dependent representation suggests a general recipe: any stochastic equation whose solution condition can be expressed as a measurable function of the path and the joint law of the path and noise should fall under the same abstract equivalence, so the framework may be adapted to other solution notions that fit the compatibility structure.","The Polish path-space assumption marks a likely boundary: solution spaces such as spaces of weakly continuous paths with a non-metrizable weak topology, or non-Polish Banach spaces, would need a different representation argument, since Kuratowski's theorem and measurable selections are used essentially.","A natural extension would be to replace the continuous path space $C(\\bar I_T;Z)$ by a Skorokhod space, which would allow discontinuous solutions and Lévy-type noise; the paper's compatibility argument is set up for continuous paths, so the stochastic integral representation would need to be rebuilt."],"forward_implications":["All previously known Yamada–Watanabe-type results for SPDEs, including mild solutions, the variational framework, and analytically weak solutions, become special cases of a single theorem.","The reverse direction of the equivalence, from strong existence plus joint weak uniqueness to pathwise uniqueness, now applies to settings it was not available for: SPDEs with time-dependent operators, quasilinear SPDEs outside the variational framework, and semilinear SPDEs in critical spaces with transport noise.","The measurable representation $I$ of the stochastic integral gives a canonical, law-dependent version of the integral in infinite dimensions, usable independently of Yamada–Watanabe theory.","When the hypotheses hold for every initial law in a class $\\mathcal{M}$, Corollary 3.15 upgrades the solution map to a single map $F$ that does not depend on the initial law."],"supporting_citations":[{"why":"Supplies the abstract Yamada–Watanabe–Engelbert theorem for compatible solutions that the paper specializes to SPDEs.","marker":"[31]"},{"why":"Earlier version of the abstract theorem whose lemmas on measurable representations and convexity structure are used in the proof.","marker":"[30]"},{"why":"Provides the Blackwell–Dubins Borel representation used to couple weak solutions in the proof.","marker":"[7]"},{"why":"Gives the functional representation for limits in probability used to construct the measurable stochastic integral.","marker":"[26]"},{"why":"Characterizes stochastic integrability in UMD spaces, defining the UMD case of the integral representation.","marker":"[34]"},{"why":"Survey of stochastic integration in Banach spaces used for martingale type 2 and UMD facts.","marker":"[33]"},{"why":"Provides the mild-solution Yamada–Watanabe theorem that this paper extends and reproves.","marker":"[38]"},{"why":"Provides the variational-framework Yamada–Watanabe theorem that this paper extends and reproves.","marker":"[42]"},{"why":"Provides the analytically weak solution Yamada–Watanabe theorem that this paper extends and reproves.","marker":"[29]"}],"fun_headline_variants":["Weak existence plus uniqueness gives strong solution map for SPDEs","Theorem unifies mild, weak, and analytic strong SPDE solutions","Yamada-Watanabe-Engelbert proved for Banach-valued SPDEs","Measurable solution map from pathwise uniqueness in Banach SPDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The path space $B$ in which solutions are required to live must be Polish and embed continuously into the space of continuous $Z$-valued paths, so that Kuratowski's theorem and measurable selection arguments apply.","fun_headline_variants_meta":{"raw":{"variants":["Weak existence plus uniqueness gives strong solution map for SPDEs","Theorem unifies mild, weak, and analytic strong SPDE solutions","Yamada-Watanabe-Engelbert proved for Banach-valued SPDEs","Measurable solution map from pathwise uniqueness in Banach SPDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000793,"raw_usage":{"total_tokens":3592,"prompt_tokens":1146,"completion_tokens":2446,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":762,"completion_tokens_details":{"reasoning_tokens":2379}},"tokens_in":762,"tokens_out":2446,"duration_ms":16678,"temperature":1.0,"reasoning_tokens":2379,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T19:52:48.644949+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the analytically weak case in a Banach space that is neither UMD nor of martingale type 2, such as $L^1$, with a nontrivial cylindrical Wiener noise: the paper asserts the equivalence holds there, and a failure of pathwise uniqueness or of the strong-solution-map conclusion in such an example would refute the claim.","supporting_citations":[{"cited_title":"Weak and strong solutions of general stochastic models","cited_arxiv_id":null,"evidence_quote":"Supplies the abstract Yamada–Watanabe–Engelbert theorem for compatible solutions that the paper specializes to SPDEs."},{"cited_title":"The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities","cited_arxiv_id":null,"evidence_quote":"Earlier version of the abstract theorem whose lemmas on measurable representations and convexity structure are used in the proof."},{"cited_title":"An extension of Skorohod’s almost sure representation theorem","cited_arxiv_id":null,"evidence_quote":"Provides the Blackwell–Dubins Borel representation used to couple weak solutions in the proof."},{"cited_title":"Kallenberg","cited_arxiv_id":null,"evidence_quote":"Gives the functional representation for limits in 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