{"id":"bc98fd70-310d-492c-a665-b5d69c793264","arxiv_id":"2608.13392","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A unified weighted Besov theory on homogeneous Lie groups with a multiscale characterization is developed and applied to prove well-posedness of parabolic Anderson models in the Young and first singular regimes.","lead":"This paper builds weighted Besov spaces, a way to measure the smoothness of very rough functions, on general homogeneous Lie groups, and proves a practical wavelet-like description of them. It then uses this toolbox to prove well-posedness of parabolic Anderson random heat equations on these groups for rough noise and rough initial data, including Dirac deltas.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Young multiplication proof (Theorem 2.15) uses an invalid convolution–pointwise-product exchange; since Theorem 3.3 and both PAM theorems rely on it, the central claims are not established as written.","rationale":"The reader's explicit weakest assumption is the deferred construction of the test-function pair (Assumption 2.7), but their rationale also flags the invalid algebraic step in Theorem 2.15. My independent reading confirms that the Young multiplication proof has a concrete, verifiable flaw: convolution and pointwise multiplication do not commute, and the telescoping recurrence that defines the product's multiscale components relies on exactly that exchange. This is the most load-bearing concern because Theorem 2.15 is not an isolated result; it is used directly in Theorem 3.3, the fixed-point theorem, which in turn yields both parabolic Anderson well-posedness theorems. The Assumption 2.7 issue is real but less decisive: the construction is deferred to prior work, and even if one accepted that reference, the Young proof would still fail as written. I therefore do not change the reader's REJECT verdict. The concern is about the validity of the argument in the submitted manuscript, not about the authors' integrity or about disagreement with external consensus. A concrete one-dimensional check can settle whether the disputed identity is genuinely invalid; if it is, the proof must be repaired before the applications can be accepted.","tokens_in":34361,"tokens_out":9383,"duration_ms":84120,"concrete_test":"On G = R, take r = 2 and a smooth compactly supported ρ with ∫ρ = 1 and vanishing first moment, so Assumption 2.7 holds in the abelian case. Set f(x) = e^x and ξ_{m+1}(x) = 1. Then ξ_{m+1} * \\tildeρ^{(m)} = 1, so f(ξ_{m+1} * \\tildeρ^{(m)}) = f(x). But (fξ_{m+1}) * \\tildeρ^{(m)}(x) = ∫ e^y 2^m ρ(2^m(x-y)) dy, which is not equal to e^x for non-constant ρ. This directly disproves the substitution used in the proof of Theorem 2.15. If the authors maintain the identity, they must exhibit the missing commutator term ∫(f(y)-f(x))ξ_{m+1}(y)ρ(x^{-1}y)dy and show it vanishes or is absorbed by the subsequent estimates.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 2.15 contains an invalid exchange of convolution and pointwise multiplication. In verifying that F_{m+1} * \\tildeρ^{(m)} = F_m, the displayed computation treats (f(ξ)_{m+1}) * \\tildeρ^{(m)} as if it were f((ξ)_{m+1} * \\tildeρ^{(m)}) = fξ_m. This is false in general: (fξ_{m+1}) * ρ(x) = ∫ f(y) ξ_{m+1}(y) ρ(x^{-1}y) dy differs from f(x) ∫ ξ_{m+1}(y) ρ(x^{-1}y) dy by the term ∫ (f(y)-f(x)) ξ_{m+1}(y) ρ(x^{-1}y) dy. The recurrence (2.28) is precisely what makes the series represent each multiscale piece of the product, and all subsequent bounds (2.29)–(2.31) estimate terms under that recurrence. Without the recurrence, Theorem 2.15 is not proven. The gap is load-bearing because Theorem 3.3 invokes Theorem 2.15 at (3.12) to estimate X^I u_s · W^{(I)}_{t,s}, and Theorems 1.3 and 1.6 are derived from Theorem 3.3. The paper therefore does not establish the Young multiplication result on which the SPDE applications rest, even if the statement itself may be repairable by a different argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an intrinsic theory of weighted, inhomogeneous Besov spaces on arbitrary homogeneous Lie groups. The definition uses localized test functions, and the central result (Theorem 2.9) asserts an equivalent multiscale characterization built from a single compactly supported test-function pair (φ,ρ). From this characterization the paper derives Besov embeddings, a Taylor-remainder characterization, a Young-type product theorem (Theorem 2.15), Schauder estimates for convolution semigroups, and a weighted Kolmogorov criterion. These tools are then applied to parabolic Anderson-type equations associated with positive Rockland operators: a Young-regime well-posedness theorem for space-time noise and a first-singular-regime result for purely spatial noise via a Cole–Hopf transform. The paper claims to provide a group-uniform, largely self-contained foundation for singular SPDEs on homogeneous Lie groups.","tokens_in":34608,"tokens_out":12530,"duration_ms":108278,"significance":"If the main results were fully established, the paper would be a valuable contribution: it would unify and extend Besov-space techniques from the Heisenberg group and Euclidean settings to general homogeneous Lie groups without representation theory, and it would provide explicit well-posedness regimes for parabolic Anderson models with rough initial data. The organizational idea is attractive, and the multiscale characterization, Schauder estimates, sewing lemma, and the Cole–Hopf application are worked out in considerable detail. However, the central Young multiplication theorem contains a fundamental algebraic error in its proof, and all SPDE applications depend on that theorem. The paper also delegates the existence of the test-function pair in Assumption 2.7 to previous work without reproducing or precisely stating the construction, making the foundational layer conditional on an external input. As written, the central claims are not established.","major_comments":[{"comment":"The verification that the series defining F_m satisfies the recurrence F_{m+1} * ρ̃^{(m)} = F_m is invalid. In the displayed computation after Eq. (2.28), the term (f(ξ)_{m+1}) * ρ̃^{(m)} is replaced by (f(ξ)_m) * ρ̃^{(m)}, and later (f(ξ)_m) * ρ̃^{(m)} is identified with f(ξ)_m. These steps effectively use the identity (f ξ_{m+1}) * ρ̃^{(m)} = f (ξ_{m+1} * ρ̃^{(m)}), which is false for nonconstant f. With ξ_m = ξ * φ̃^{(m)}, it may be true that ξ_{m+1} * ρ̃^{(m)} = ξ_m, but (f ξ_{m+1}) * ρ̃^{(m)}(x) = ∫ f(y) ξ_{m+1}(y) ρ̃^{(m)}(x^{-1}y) dy is not equal to f(x) ξ_m(x); the difference is a commutator term of exactly the type the correction series in (2.28) is intended to encode. Without the recurrence, the subsequent bounds (2.29)–(2.31) do not establish convergence of the series to the product or the desired Besov estimate. This gap is load-bearing: Theorem 3.3 invokes Theorem 2.15 at (3.12), and Theorems 1.3 and 1.6 are consequences of Theorem 3.3. Thus the Young multiplication result and the SPDE applications are not proved as written.","section":"Section 2.4, Theorem 2.15 and Eq. (2.28)"},{"comment":"Theorem 2.9, and therefore every result in the paper, is conditional on the existence of a pair (φ,ρ) satisfying the support, moment, and convergence properties in Assumption 2.7. The manuscript does not prove this existence: Remark 2.8 only states that the construction is elementary and refers to [MS25, Lem. 3.10]. Since the paper advertises itself as intrinsic and largely self-contained, and since the entire multiscale characterization collapses if no such pair exists for a given homogeneous Lie group, the paper should either reproduce the construction or state precisely the conditions on r, R, and the group under which such a pair exists. As it stands, the foundational input is unverified within the manuscript.","section":"Assumption 2.7 and Remark 2.8"}],"minor_comments":[{"comment":"The expression \"sup_{φ∈B_{-α'}⌉}\" contains a stray closing bracket symbol that should be removed.","section":"Section 2.1, display after Eq. (2.13)"},{"comment":"The notation ρ^{(m,m-1)}_z := 1 is introduced without explanation; please define it explicitly in terms of the convolution convention used for ρ^{(n,m)}.","section":"Section 2.4, proof of Theorem 2.15"},{"comment":"The mollified noise ξ_ε = ξ * ρ_ε is used before ρ_ε is defined; please specify that ρ_ε(x) = ε^{-|s|} ρ(ε^{-1} · x).","section":"Theorem 1.6"},{"comment":"The assertion that both terms in inequality (2.8) vanish along a subsequence is terse. The second term is O(λ), and the first term should be quantified as O(λ^{α-d(I)}) using the Besov norm; please make this explicit.","section":"Lemma 2.5"}],"recommendation":"reject","confidential_remarks":"The paper has a strong conceptual framework and the applications are interesting, but the proof of the main Young multiplication theorem contains a false convolution–pointwise-product exchange that invalidates the central functional-analytic result. Since Theorem 3.3 and both parabolic Anderson theorems depend on Theorem 2.15, the paper's central claims are unproven as written. If the authors can supply a correct proof of Theorem 2.15 and provide the missing construction behind Assumption 2.7, the paper would be worth reconsidering."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is an ambitious paper: it builds an intrinsic multiscale characterization of weighted Besov spaces on arbitrary homogeneous Lie groups and then uses it to extend parabolic Anderson well-posedness results to Rockland operators with singular initial data. The framework is attractive, and the multiscale theorem (2.9) is a real contribution, including in the Euclidean case for p not equal to infinity. The embeddings, Taylor remainder, Schauder estimates, and Kolmogorov criterion are organized cleanly, and the applications genuinely go beyond the Heisenberg/Carnot settings.\n\nBut there is a load-bearing flaw. In the proof of Theorem 2.15 (Young multiplication, alpha <= 0), the author proves F_{m+1} * rho~(m) = F_m by replacing (f xi_{m+1}) * rho~(m) with f (xi_{m+1} * rho~(m)) = f xi_m. That exchange is false for nonconstant f: convolution averages f against the mollifier, so (f xi_{m+1}) * rho~(m) is not f times a mollified xi. The displayed identity (2.28) and the recurrence that follows are exactly what make the series representation work; without them, the estimates for the product do not follow. Since Theorem 3.3 invokes Theorem 2.15 at (3.12) and both PAM theorems are downstream, the central claims are not established as written. The error looks repairable, likely by a more careful paraproduct-type expansion that keeps track of the commutator, but it is not a cosmetic gap.\n\nTwo other weaknesses, in proportion. The proof of Claim 3.7 (convergence of the renormalized quadratic term) is a sketch with a reference to standard arguments; for a result that is one of the two advertised applications, that is thinner than ideal. And Assumption 2.7 inherits the existence of the test-function pair from the author's prior work [MS25] with only a pointer; that is acceptable practice, but it means the whole multiscale enterprise rests on an external construction the reader cannot check quickly.\n\nWhat is genuinely good: the multiscale characterization and the machine built around it. The Besov parts are detailed and plausible; the Schauder estimates and the sewing lemma look like useful tools. The paper is not one of free parameters or fitted results; the regimes are explicit.\n\nFor peer review: yes, I would send this to a serious referee. The framework is significant enough, and the flaw, while load-bearing, is identifiable and likely fixable. A referee should demand a corrected proof of Theorem 2.15 and a fuller proof of Claim 3.7, but the paper deserves the attention. If the Young step cannot be repaired, the applications fall; if it can, the rest is in good shape.","headline":"The paper's framework is genuinely useful, but the proof of the Young multiplication theorem contains an invalid convolution-pointwise-product exchange that the SPDE applications rely on; it deserves a serious referee but not acceptance as written.","tokens_in":35179,"tokens_out":3577,"would_cite":false,"duration_ms":33652,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E35","35H20","60H15","43A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"One pair of compactly supported test functions carries the entire weighted Besov calculus on every homogeneous Lie group, and the same machinery proves well-posedness for parabolic Anderson models in the Young and first singular regimes.","keywords":["Besov spaces","homogeneous Lie groups","weighted function spaces","multiscale characterization","parabolic Anderson model","Rockland operators","Cole-Hopf transform","singular SPDEs"],"falsifier":"Take a concrete homogeneous Lie group not covered by Heisenberg-specific arguments, for instance the Engel group, construct the pair $(\\varphi,\\rho)$ explicitly from the cited lemma, and compare the Definition 5 Besov norm of $\\delta_e$ with the multiscale norm (2.9) for $\\alpha=-1$ and $p=q=2$; Theorem 2.9 predicts a bounded ratio independent of the pair. If the ratio is unbounded, or if the limit $\\rho^{(n,m)}\\to\\varphi^{(n)}$ fails in $C^\\infty_c$ rather than merely in distributions, the central equivalence is false.","tokens_in":41,"feed_emoji":"","tokens_out":15249,"duration_ms":882554,"temperature":0.7,"pith_summary":"This paper sets out to show that a single pair of compactly supported smooth test functions carries the entire weighted Besov theory on every homogeneous Lie group, with no case-by-case group structure and no representation theory. Its central theorem, Theorem 2.9, asserts that the Besov norm defined through localised test functions is equivalent, for regularities $\\alpha$ outside the dilation spectrum, to a wavelet-like multiscale norm built from convolutions at dyadic scales. From that equivalence the paper derives, in one framework, Besov embeddings, a Taylor-remainder characterisation, Young-type multiplication, Schauder estimates for heat semigroups of a positive Rockland operator, and a Kolmogorov criterion for random distributions. The payoff is two well-posedness theorems for parabolic Anderson models: space-time noise in the Young regime with initial data as rough as Dirac masses, and purely spatial noise in the first singular regime via a Cole-Hopf transform adapted to Rockland operators. A sympathetic reader should take the paper's claim to be that one mechanism, the test-function pair of Assumption 2.7, organizes all of these results.","feed_headline":"One pair of test functions gives Besov theory on any homogeneous group","feed_subtitle":"No representation theory and no group-by-group arguments are needed.","key_machinery":"The load-bearing object is the test-function pair $(\\varphi,\\rho)$ of Assumption 2.7: both are compactly supported and smooth, $\\rho$ has vanishing moments up to order $R$, and the repeated convolutions $\\rho^{(n,m)}=\\rho^{(n)}*\\cdots*\\rho^{(m)}$ converge to $\\varphi^{(n)}$ in $C^\\infty_c$. This pair generates the dyadic sequence $\\xi_n=\\xi*\\tilde\\varphi^{(n)}$, and Theorem 2.9 expresses the Besov norm of $\\xi$ as an $\\ell^q(L^p)$ sum of these pieces, which is what makes the later estimates look Euclidean. The other essential mechanism is the integral-form Taylor theorem (Theorem 1.10), which represents remainders by integrable measures and is used for the $\\alpha>0$ direction, the Taylor-remainder norm of Theorem 2.13, and the Young multiplication theorem 2.15. For the SPDE applications, the mild sewing lemma (Lemma 3.1) handles time-dependent weighted Banach spaces and increments that blow up near $t=0$, while the Cole-Hopf identities of Section 3.3 transform the Anderson equation into a Young equation for $w=e^{-v}u$.","core_discovery":"The central discovery is Theorem 2.9: for $\\alpha \\notin \\Delta$, where $\\Delta$ is the dilation spectrum, the weighted inhomogeneous Besov norm of Definition 5 is equivalent to the multiscale quantity (2.9) when $\\alpha<0$ and to (2.10) when $\\alpha>0$, provided the pair $(\\varphi,\\rho)$ of compactly supported smooth functions satisfies Assumption 2.7. The paper proves the equivalence in both directions, then uses it to obtain embeddings, the Taylor-remainder norm, Young multiplication, convolution estimates for singular kernels, and the Kolmogorov criterion. On the SPDE side, Theorem 3.3 turns these estimates into a fixed-point theorem for the mild equation, Theorem 1.3 gives global well-posedness in the Young regime for singular initial data, and Theorem 1.6 establishes well-posedness in the first singular regime for purely spatial noise, with the renormalised solution independent of the mollifier and obtained from a variant of the Cole-Hopf transform for Rockland operators.","pith_inferences":["Because the proof of Theorem 2.9 avoids the group's representation theory, the same two-function characterization ought to transfer to any homogeneous space admitting a metric dilation and an integral Taylor theorem; the paper does not explicitly claim this.","The Cole-Hopf transformation for Rockland operators writes the transformed equation as a sum of Young products whose coefficients are Bell polynomials in derivatives of $v$; continuing the same transformation below $\\zeta_\\star$ would require renormalising cubic and higher terms, so this construction is a natural route into the next singular regime.","The sewing lemma's time-dependent weighted norms were built for spatially unbounded noises on $G$; a testable consequence is that the same lemma should yield pathwise well-posedness for other singular SPDEs on non-compact groups, provided their heat kernels satisfy bounds like (2.32)."],"forward_implications":["Besov embeddings on homogeneous Lie groups, including the Sobolev-type inequality with homogeneous dimension $|s|$, follow from one dyadic argument once Theorem 2.9 is available (Corollary 2.12).","Pointwise multiplication extends to weighted Besov spaces whenever $\\alpha+\\beta>0$, so the paraproduct estimates needed in Young-regime SPDEs do not require a group-specific construction (Theorem 2.15).","Heat semigroups of positive Rockland operators satisfy sharp Schauder estimates in these spaces, giving the time regularisation used in the fixed-point proof (Proposition 2.19).","Parabolic Anderson equations on arbitrary homogeneous Lie groups are globally well posed in the Young regime for stationary space-time noise with $\\alpha/m+H>1/2$, even with initial data as rough as $\\delta_e$ (Theorem 1.3).","For purely spatial Gaussian noise of regularity $\\zeta$ in the interval determined by $\\zeta_\\star$, the mollified, renormalised Anderson equations converge in probability to a limit that is independent of the mollifier (Theorem 1.6)."],"supporting_citations":[{"why":"Supplies the existence of the test-function pair of Assumption 2.7 and the integral-form Taylor theorem (Theorem 1.10) on which Theorem 2.9 leans.","marker":"[MS25]"},{"why":"Provides the Heisenberg-group weighted Besov and parabolic Anderson theory that Theorem 1.3 extends to general homogeneous Lie groups and singular initial data.","marker":"[BCH+25]"},{"why":"Supplies the reconstruction and sewing arguments, including the presentation of singular increments, adapted by the proof of Theorem 2.9 and Lemma 3.1.","marker":"[FH20]"},{"why":"Gives the rough evolution equation sewing lemma that Lemma 3.1 generalises to time-dependent norms and increments singular at $t=0$.","marker":"[GT10]"},{"why":"Establishes the Gaussian heat-kernel bounds for semigroups generated by positive Rockland operators, which verify Assumption 2.16 behind the Schauder estimates.","marker":"[DHZ94]"},{"why":"Provides weighted subcoercive estimates for Rockland-type operators, used to justify the kernel bounds and convolution estimates of Section 2.5.","marker":"[tR98]"},{"why":"Supplies standard facts about homogeneous groups, Taylor polynomials, and convolution inequalities used throughout Sections 1 and 2.","marker":"[FS82]"},{"why":"Supplies the regularity-structures power counting that identifies the Young and singular regimes reproduced here with homogeneous dimension and degree.","marker":"[Hai14]"},{"why":"Provides the Euclidean Cole-Hopf construction for the continuum parabolic Anderson model that the Rockland-operator variant in Theorem 1.6 adapts.","marker":"[HL15]"}],"fun_headline_variants":["Two test functions unify Besov theory on all homogeneous groups","No group-specific arguments: Besov theory on any homogeneous group","One norm equivalence yields Besov embeddings and Anderson well-posedness","Besov theory on all homogeneous groups via a single test pair"],"cache_read_input_tokens":37248,"weakest_assumption_plain":"The whole construction rests on Assumption 2.7: on every homogeneous Lie group in question there must exist a compactly supported smooth pair $(\\varphi,\\rho)$ with $\\rho$ having vanishing moments to order $R$ and with $\\rho^{(n,m)}\\to\\varphi^{(n)}$, a fact the paper imports from previous work rather than proving here; if such a pair fails for some admissible group, the multiscale characterisation and every application built on it collapses.","fun_headline_variants_meta":{"raw":{"variants":["Two test functions unify Besov theory on all homogeneous groups","No group-specific arguments: Besov theory on any homogeneous group","One norm equivalence yields Besov embeddings and Anderson well-posedness","Besov theory on all homogeneous groups via a single test pair"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001273,"raw_usage":{"total_tokens":5194,"prompt_tokens":918,"completion_tokens":4276,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":4204}},"tokens_in":534,"tokens_out":4276,"duration_ms":33173,"temperature":1.0,"reasoning_tokens":4204,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:56:43.004736+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete homogeneous Lie group not covered by Heisenberg-specific arguments, for instance the Engel group, construct the pair $(\\varphi,\\rho)$ explicitly from the cited lemma, and compare the Definition 5 Besov norm of $\\delta_e$ with the multiscale norm (2.9) for $\\alpha=-1$ and $p=q=2$; Theorem 2.9 predicts a bounded ratio independent of the pair. If the ratio is unbounded, or if the limit $\\rho^{(n,m)}\\to\\varphi^{(n)}$ fails in $C^\\infty_c$ rather than merely in distributions, the central equivalence is false.","supporting_citations":[],"review_version":1}