{"id":"95c55821-7576-496e-b425-c627f61f0ac6","arxiv_id":"2507.06851","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The paper proves convergence of renormalised models in regularity structures for variable coefficient singular SPDEs across full subcritical regimes, with renormalisation functions depending only on a finite jet of the coefficient field.","lead":"This paper proves that the renormalisation machinery behind Hairer's regularity structures, used to make sense of very noisy stochastic PDEs, still works when the main differential operator has coefficients that vary in space and time instead of being constant.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.14 presumes a uniform-in-kernel BPHZ estimate from [CH16, HS24, BH23] that Theorem 2.12 does not state as an hypothesis; if the cited results only cover fixed kernels, the lifting step is unproved.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: Lemma 5.14 assumes a uniform-in-kernel form of the external BPHZ estimates. This is indeed the most critical point because it is the bridge that lets the paper lift stochastic estimates from the scalar-valued theory to the infinite-dimensional structure TBan, and from there to the final model on Teq. The paper is otherwise architecturally detailed and internally coherent: the construction of TBan via universal properties, the preparation-map formalism, and the pointed-modelled-distribution transfer are all developed at length. But the reliance on [CH16, HS24, BH23] is not merely a citation for a known result; Lemma 5.14 requires a strictly stronger statement than those papers appear to state. The text's own caveat 'so long as those scalar valued counterparts are assumed to be suitably uniform' confirms that this is a real external premise. If the premise fails, the main theorem is not proved, but the nature of the gap is a missing verification rather than a fatal contradiction. A careful check of the cited proofs, with explicit kernel dependence, could settle the point. Because this is the same concern the reader already identified and the conditional verdict is appropriate, I do not recommend changing the verdict.","tokens_in":69006,"tokens_out":5836,"duration_ms":67577,"concrete_test":"Re-derive the scalar BPHZ estimate of [HS24, Theorem 2.4] (or [CH16, Theorem 1.1] / [BH23, Theorem 2.6]) on the auxiliary structures Tτ while tracking the dependence on the kernel assignment Aτ. Verify whether the bound sup_{Aτ} E^{1/p}|Π̂τ_x τ* (φ^λ_x)|^p / ∏_e ||Aτ(l(e))|| < ∞ holds with a constant independent of Aτ, and similarly for the difference estimate used in (5.7). If the constant depends on Aτ beyond its norm, or if only fixed kernels are treated, Lemma 5.14 is an additional assumption and Theorem 2.12 must be amended.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 2.12 hinges on Lemma 5.14, which asserts the annealed estimate with a supremum over kernel assignments Aτ on the right-hand sides of (5.6) and (5.7). The proof of Lemma 5.14 does not establish this uniformity; the text preceding it explicitly says the estimate holds 'so long as those scalar valued counterparts are assumed to be suitably uniform in the kernel assignment and to hold on the regularity structures Tτ'. But Theorem 2.12's hypotheses only say that (Teq, ξ) satisfies the assumptions of at least one of [CH16, HS24, BH23]. Those works are formulated for a fixed translation-invariant kernel (or a specific model), and their published statements do not in an obvious way provide a supremum over all kernel assignments Aτ on the auxiliary structures Tτ with a constant independent of Aτ. The two-model version (5.7), needed for the local Lipschitz continuity of K ↦ Z, is even farther from the stated inputs, since it also requires uniformity of the difference estimate as Aτ varies. If this uniformity fails, the annealed estimates on TBan, and consequently the quenched estimates and Theorem 2.12, would require an additional argument or a modified hypothesis. This is a missing verification of an external premise rather than an internal contradiction, so it warrants a conditional acceptance pending a concrete check of the cited results.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework for constructing renormalised models in Hairer's regularity structures for variable-coefficient singular SPDEs. It introduces an auxiliary regularity structure TBan whose components are partially symmetrised projective tensor products of infinite-dimensional Banach spaces, builds admissible models from preparation maps, and constructs a BPHZ model on historic sectors. Using pointed modelled distributions, it transfers these models to the standard reduced structure Teq, yielding convergence of mollified renormalised models (Theorem 2.12), Lipschitz dependence on the kernel assignment, and an explicit form of the state-space-dependent preparation map. Under Assumption 2.15, the renormalisation functions depend only on a finite jet of the coefficient field (Corollary 2.16), and the paper verifies the assumption for second-order parabolic operators via a detailed heat-kernel decomposition in Section 9. The paper is explicit that the noise itself remains translation invariant in law (Remark 2.14).","tokens_in":69242,"tokens_out":7900,"duration_ms":90924,"significance":"If correct, this is a significant contribution: it supplies a missing variable-coefficient BPHZ input, generalising the scope of [CH16, HS24, BH23] beyond translation-invariant coefficients, and it provides a locality mechanism for counterterms. The universal-property construction of TBan is elegant, the analytic framework of pointed modelled distributions is developed in useful generality, and the heat-kernel decomposition in Section 9 is a nontrivial verification. The paper is also unusually explicit about its hypotheses and limitations. However, the central stochastic lifting argument inherits an external uniformity assumption that is not stated as a hypothesis of Theorem 2.12, and several load-bearing analytic lemmas are only sketched. The result is credible but needs these gaps closed.","major_comments":[{"comment":"The annealed estimates (5.6)-(5.7) assert bounds whose right-hand side contains a supremum over all kernel assignments Aτ on the auxiliary scalar structures Tτ. The proof of Lemma 5.14 obtains the inequality by taking that supremum as the normalising constant for the given induced assignment; it does not prove that the supremum is finite or bounded uniformly in the relevant data. The text before the lemma explicitly makes this uniform-in-kernel property an assumption ('so long as those scalar valued counterparts are assumed to be suitably uniform in the kernel assignment'), but Theorem 2.12's hypotheses only say that (Teq, ξ) satisfies the assumptions of one of [CH16, HS24, BH23]. If those cited theorems are only stated for fixed kernels, the annealed estimate, and hence the quenched estimates and Theorem 2.12, are incomplete. The two-model version (5.7), needed for local Lipschitz continuity of K ↦ Z, additionally requires uniformity of difference estimates for two noise assignments, which is even further from the stated inputs. Please either verify this uniformity from the cited results, add it as an explicit hypothesis, or supply a proof.","section":"§5.1, Lemma 5.14 and Theorem 2.12"},{"comment":"The planted-tree case of Lemma 4.19 is dismissed as an instance of [Hai14, Theorem 5.14] with a 'minor adaptation' for two different kernels, and the details are omitted. This lemma is load-bearing: it provides the Γ-bounds used in Lemma 4.20, in the quenched estimates of Lemma 5.23, and in the continuity of the model as a function of the kernel assignment. Since the components of TBan are infinite-dimensional and the two-model comparison involves different kernel assignments as well as different noise assignments, the extension is not a routine re-reading of the scalar theorem. Please provide a complete proof or a precise reference that covers this situation.","section":"§4.2, Lemma 4.19"},{"comment":"Theorem 2.12 is stated without explicitly listing the translation invariance of the noise among its hypotheses, even though the proof relies on it and Remark 2.14 later acknowledges it. This is not a mathematical error, but it makes the statement of the main theorem misleading. The assumption should be moved into the theorem statement or the abstract should be adjusted so that the reader is not led to believe the result covers non-translation-invariant noise.","section":"§2.1, Remark 2.14 and Theorem 2.12"}],"minor_comments":[{"comment":"The name 'Kupianen' in the introduction appears to be a typo for 'Kupiainen' (reference [Kup16]).","section":"§1, references"},{"comment":"The word 'inpired' should be 'inspired'.","section":"§2.2, Strategy of Proof"},{"comment":"In the displayed formula in the proof of Lemma 5.13, the expression '⊗ ¯w 〈 ¯www 〉' contains a typo; it should read '⊗ ¯w 〈 ¯w 〉' or the intended symbol should be corrected.","section":"§5.1, Lemma 5.13 proof"},{"comment":"The verification of Assumption 2.15 for second-order parabolic operators is stated under Hölder regularity conditions that the authors say are 'probably not optimal'. This limitation should be stated more prominently in the introduction or near Corollary 2.16, since the locality conclusion is conditional on it.","section":"§9, Assumption 2.15"}],"recommendation":"major_revision","confidential_remarks":"The paper is strong and likely publishable if the uniform-in-kernel estimates issue is resolved. In particular, the authors should either point to exact theorems in [CH16, HS24, BH23] that provide the supremum over kernel assignments in (5.6)-(5.7), or add this as an explicit hypothesis to Theorem 2.12. The two-model version of the estimate also needs a concrete justification. The proof of Lemma 4.19 should be completed rather than left as a 'minor adaptation'."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is the missing variable-coefficient BPHZ input for the regularity structures program. Theorem 2.12 gives convergence of renormalised models for variable coefficient singular SPDEs in full subcritical regimes, and the mechanism is an infinite-dimensional regularity structure TBan that freezes coefficients at the model level. Corollary 2.16 gives local renormalisation functions depending on the coefficient field only through a finite jet, with the kernel decomposition checked for second order parabolic operators in Section 9. That is genuinely new and, as far as I can audit, coherent. The paper is also unusually explicit about what it assumes and what it delegates.\n\nWhat is good: the freezing strategy is clean, and the transfer from TBan to the usual scalar-valued structure via pointed modelled distributions is a real methodological step forward. The authors deliberately do not tie the result to one probabilistic technique; they take BPHZ estimates from any of CH16, HS24, BH23 as input. They also state limitations plainly, including Remark 2.14 conceding that the noise remains translation invariant in law. That matters, but it is honest.\n\nSoft spots, in proportion: the main theorem is proved on finite historic sectors, and convergence on the full structure is delegated to standard post-processing. Several load-bearing proofs are sketched or omitted: Lemma 4.19, Proposition 4.17, Lemma 7.3. These are probably repairable, but a referee needs to see them.\n\nThe concern I would push hardest is Lemma 5.14. The annealed estimates on TBan are reduced to scalar BPHZ estimates on auxiliary structures Tτ with a supremum over kernel assignments Aτ. The text says the scalar estimates must be “suitably uniform in the kernel assignment,” but Theorem 2.12’s hypotheses only require the pair to satisfy assumptions of one of CH16, HS24, BH23. Those papers are formulated for fixed kernels. If the uniform-in-kernel version is not already present in them, the lifting step needs a separate argument or a modified hypothesis. The two-model version, used for local Lipschitz continuity, makes the same demand for differences. This is not an internal contradiction; it is a missing verification of an external premise, and it is the main thing I would want checked case by case.\n\nBottom line: this deserves serious peer review, not desk rejection. It fills a real gap and the architecture is sound. Send it to a careful referee who can verify Lemma 5.14 against the cited inputs and request full proofs of the sketched lemmas.","headline":"First variable-coefficient BPHZ theorem for regularity structures; substantive and mostly honest, but the lifting step leans on an unverified uniformity-in-kernel premise that a referee must check.","tokens_in":69925,"tokens_out":2353,"would_cite":true,"duration_ms":29369,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R60","60H17","60H15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Variable-coefficient singular SPDEs get convergent renormalised models.","keywords":["regularity structures","renormalised models","singular SPDEs","variable coefficients","BPHZ renormalisation","parabolic operators","stochastic PDE estimates","locality of counterterms"],"falsifier":"Find a kernel assignment and a noise satisfying one of the input hypotheses for which the uniform supremum over $A^\\tau$ on the right-hand side of (5.6) or (5.7) diverges for some tree in a historic sector; that would break the annealed estimates and with them Theorem 2.12. Equivalently, in a concrete second-order parabolic example such as $\\partial_t u = \\partial_i(a^{ij}(x)\\partial_j u) + u\\,\\xi$ with $a^{ij}(x) = \\delta^{ij} + \\varepsilon \\eta^{ij}(x)$, compute the BPHZ renormalisation function for the simplest negative-degree tree and check whether its dependence on $\\eta$ factors through the finite jet $(\\eta(x), \\nabla\\eta(x))$ prescribed by Corollary 2.16.","tokens_in":68660,"feed_emoji":"🧮","tokens_out":9867,"duration_ms":100376,"temperature":0.7,"pith_summary":"This paper closes the main gap between constant-coefficient and variable-coefficient singular SPDEs in the regularity structures approach: convergence of renormalised models as the ultraviolet cut-off is removed. The central result, Theorem 2.12, says that for any finite historic sector, whenever the noise satisfies the assumptions of at least one of the three earlier constant-coefficient BPHZ theorems, every kernel assignment admits a sequence of state-space-dependent preparation maps whose renormalised models converge in every $L^p$, with the limiting model locally Lipschitz continuous in the coefficient field. Because translation invariance is not available, counterterms must be functions of the space-time point rather than constants, and Corollary 2.16 shows they can be chosen local: the renormalisation functions depend on the point only through a finite-order jet of the coefficient field, verified for second-order parabolic operators in Section 9. The upshot is that the variable-coefficient case offers no fundamental obstruction: the full subcritical regime is in reach whenever the constant-coefficient BPHZ estimates are available.","feed_headline":"Variable-coefficient singular SPDEs get convergent renormalised models","feed_subtitle":"Counterterms become local functions of the coefficient field, opening the full subcritical regime beyond translation-invariant equations.","key_machinery":"The load-bearing object is an auxiliary regularity structure $T_{\\mathrm{Ban}}$ whose homogeneous components are partially symmetrised projective tensor products of infinite-dimensional Banach spaces: instead of a copy of $\\mathbb{R}$ attached to each tree edge, each edge carries a space of kernels, so abstract integration can realise the whole family of kernels $K^z$ at once. The BPHZ preparation map on $T_{\\mathrm{Ban}}$ is of the form $(\\hat{\\ell}\\otimes \\mathrm{id})\\Delta^-_r$, with $\\Delta^-_r$ the rooted extraction coproduct; stochastic estimates are lifted from scalar-valued structures $T^\\tau$ constructed for each tree, and annealed bounds are converted to quenched bounds by a Kolmogorov argument. Pointed modelled distributions then transfer the model back to the usual reduced structure $T_{\\mathrm{eq}}$, and an algebraic comparison between $\\Delta^-_r$ and an auxiliary coproduct identifies the preparation map explicitly, with $\\ell^\\varepsilon_z$ depending on $z$ only through the finite jet of the kernel assignment.","core_discovery":"On the paper's own terms, the discovery is that renormalisation in a non-translation-invariant setting can be performed by freezing the coefficient field at the space-time point while still obtaining a genuine model on the ordinary scalar regularity structure. Precisely, for each finite historic sector $\\langle B\\rangle$ and each kernel assignment $K$—a choice of regularising kernels for each kernel type, allowed to vary continuously with the space-time point—there is a sequence of preparation maps $P^\\varepsilon(z,\\tau) = (\\ell^\\varepsilon_z \\otimes \\mathrm{id})\\Delta^-_r \\tau$ such that the resulting renormalised models converge in $L^p$ to a limit $Z$ that does not depend on the mollifier and depends locally Lipschitz-continuously on $K$; the functional $\\ell^\\varepsilon_z$ depends on $z$ only through the finite collection $(\\partial^k K^z)_{|k|<m}$. When the kernel assignment comes from the Green's function of a second-order parabolic operator with Hölder coefficients, this forces the counterterms in the renormalised equation to be local functions of the coefficient field and finitely many of its derivatives at each point. Combined with the existing analytic and algebraic machinery, this yields local-in-time well-posedness for a wide class of variable-coefficient subcritical singular SPDEs in their full subcritical regimes.","pith_inferences":["Editorial inference: the same freezing-at-the-point strategy is a natural template for geometric settings; scalar SPDEs on manifolds whose universal cover is $\\mathbb{R}^d$ lift to variable-coefficient equations, so this result suggests the remaining ingredient for a geometric BPHZ theorem is the analogue of the heat-kernel decomposition.","Editorial inference: the explicit formula $P^\\varepsilon = (\\ell^\\varepsilon_z \\otimes \\mathrm{id})\\Delta^-_r \\tau$ expresses the variable-coefficient counterterm through constant-coefficient renormalisation constants of auxiliary structures, which may make the renormalised equation's dependence on the coefficients computable in examples.","Editorial inference: the Hölder regularity assumed on the coefficients in Section 9 is probably not optimal; testing whether the locality proof survives with weaker coefficients would delineate the true scope of the result.","Editorial inference: the restriction to rational scaling ratios is described in the paper as technical, so a natural check is whether the wavelet-based arguments can be replaced by a scale-blind argument to remove it."],"forward_implications":["If correct, Theorem 2.12 supplies the missing BPHZ convergence input, so the standard fixed-point machinery gives local-in-time well-posedness for variable-coefficient subcritical singular SPDEs in their full subcritical regimes.","Corollary 2.16 implies the renormalised equation has a counterterm of the form $c^\\varepsilon_\\tau[(\\partial^j a(z))_{|j|<N}]$, so the counterterms are local and the renormalised equation is translation-equivariant when the coefficient field is.","The limiting model $Z$ is a locally Lipschitz continuous function of the kernel assignment and is independent of the mollifier.","Because the proof imports stochastic estimates rather than one particular probabilistic technique, the result inherits the full range of noise assumptions covered by any of the three input BPHZ theorems.","For second-order parabolic operators, Section 9's kernel decomposition verifies the locality assumption, so the theorem applies concretely to $L = \\partial_t - a^{ij}\\partial_i\\partial_j - b^i\\partial_i - c$ with Hölder coefficients."],"supporting_citations":[{"why":"Foundational source for regularity structures, admissible models, reconstruction, and the analytic Schauder machinery used throughout.","marker":"[Hai14]"},{"why":"Constructs the reduced regularity structure $T_{\\mathrm{eq}}$ from complete subcritical rules and supplies algebraic renormalisation for scalar constant-coefficient structures.","marker":"[BHZ19]"},{"why":"One of the three input BPHZ theorems supplying scalar stochastic estimates for renormalised models in the constant-coefficient setting.","marker":"[CH16]"},{"why":"Input BPHZ theorem via spectral gap estimates, and the source of pointed modelled distributions that Section 6 adapts.","marker":"[HS24]"},{"why":"Input BPHZ theorem supplying scalar stochastic estimates under a different set of noise assumptions.","marker":"[BH23]"},{"why":"Preparation maps and the coproduct $\\Delta^-_r$ used to define the BPHZ preparation map and build models recursively.","marker":"[Bru18]"},{"why":"Shows preparation-map renormalisation adapts to variable coefficients and connects preparation maps to the form of the renormalised equation.","marker":"[BB21]"},{"why":"Template for regularity structures with infinite-dimensional components and for lifting BPHZ estimates to such structures.","marker":"[GH19]"},{"why":"Provides symmetric-set tensor products used in Section 3 to make tree products and coproducts compatible with symmetries.","marker":"[CCHS22]"},{"why":"Volterra-series construction of heat kernels used in Section 9 to decompose Green's functions for parabolic operators.","marker":"[Gri04]"}],"fun_headline_variants":["Variable-coefficient SPDEs get convergent renormalised models","Local counterterms unlock full subcritical regime for SPDEs","Freeze coefficients, get convergent models for singular SPDEs","Renormalisation functions go local for variable-coefficient SPDEs","Beyond translation-invariant: SPDE models now converge"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the scalar-valued BPHZ estimates taken from the earlier constant-coefficient theorems hold uniformly in the kernel assignment on the auxiliary structures $T^\\tau$, exactly as the suprema on the right-hand side of (5.6) and (5.7) require; those earlier theorems were proved for particular constant-coefficient models, and this paper does not rederive that uniform-in-kernel form.","fun_headline_variants_meta":{"raw":{"variants":["Variable-coefficient SPDEs get convergent renormalised models","Local counterterms unlock full subcritical regime for SPDEs","Freeze coefficients, get convergent models for singular SPDEs","Renormalisation functions go local for variable-coefficient SPDEs","Beyond translation-invariant: SPDE models now converge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000325,"raw_usage":{"total_tokens":1837,"prompt_tokens":975,"completion_tokens":862,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":775}},"tokens_in":591,"tokens_out":862,"duration_ms":8882,"temperature":1.0,"reasoning_tokens":775,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:54:01.203070+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a kernel assignment and a noise satisfying one of the input hypotheses for which the uniform supremum over $A^\\tau$ on the right-hand side of (5.6) or (5.7) diverges for some tree in a historic sector; that would break the annealed estimates and with them Theorem 2.12. Equivalently, in a concrete second-order parabolic example such as $\\partial_t u = \\partial_i(a^{ij}(x)\\partial_j u) + u\\,\\xi$ with $a^{ij}(x) = \\delta^{ij} + \\varepsilon \\eta^{ij}(x)$, compute the BPHZ renormalisation function for the simplest negative-degree tree and check whether its dependence on $\\eta$ factors through the finite jet $(\\eta(x), \\nabla\\eta(x))$ prescribed by Corollary 2.16.","supporting_citations":[],"review_version":1}