{"id":"a212068f-e672-41a3-8f19-0e80b6a92f66","arxiv_id":"2607.27472","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A stochastically forced Keller–Segel system in dimensions d≥3 admits maximal local solutions in scaling-critical Besov spaces, and small data guarantees arbitrarily long existence with high probability.","lead":"This paper proves that a stochastic version of the Keller–Segel chemotaxis model has unique solutions in critical Besov spaces on tori of dimension three and higher, and that small initial data yields long-time existence with high probability. It shows that the modern 'stochastic maximal regularity' framework can handle the nonlinear chemotaxis drift in these scaling-critical spaces.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the theorem's proof verifies the framework's criticality condition directly, so the unsupported 'noise is lower order' claim in §3.1 does not affect correctness.","rationale":"I read the full manuscript and re-checked the main estimates. The proof of Theorems 3.4 and 3.5 is internally consistent: the triangular reduction for the linear operator in §4.1 is sound; the deterministic nonlinearity estimates in Lemma 4.3 follow from the Sobolev embeddings with the algebra of (3.6)–(3.10); the noise estimates in Lemmas 4.4–4.6 are carefully tuned and satisfy the Agresti–Veraar criticality condition (4.33), including the subcritical choices for g2 in the case sv∈(0,1]. The small-data argument in Lemma 5.1 and the iteration in Theorem 3.5 are also coherent, with no evident circularity after fixing the SMR constant on [0,1] and choosing T0 later. The reader's identified weakest assumption—that the noise is subcritical—is indeed asserted rather than proved in §3.1, and the cross-reference to Assumption 3.3 is inaccurate. However, this assertion is not needed for the proof: the well-posedness conclusion follows from the direct verification of (4.33), which is the actual notion of criticality used by the framework. Allowing the most singular linear noise permitted by Assumption 3.3 still satisfies those estimates. Therefore the concern does not threaten the central mathematical claim; it only affects the terminology and the heuristic justification of the critical spaces. A clarifying remark would be appropriate, but the verdict should remain conditional rather than being strengthened or weakened on mathematical grounds.","tokens_in":30563,"tokens_out":31208,"duration_ms":270252,"concrete_test":"Compute explicitly, for the worst-case linear noise g1(u,v,∇v)=∇v and g2(v)=v in the unique d=3 parameter set (p,κ,s,q,sv,qv)=(2,0,−3/2,2,1/2,2), the γ-radonifying norms ∥G(U)∥_{γ(ℓ2×ℓ2, H^{1+s,2}×H^{1+sv,2})} and check that the estimates (4.18)–(4.21) with the parameters of Theorem 3.4 satisfy (4.33). If they do, the theorem does not rely on the §3.1 heuristic and the manuscript only needs a clarifying remark; if they fail, the proof has a gap for the allowed noise class.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single passage flagged by the reader is Section 3.1, where the critical space (3.5) is derived by 'neglecting contributions of the noise terms, which we take to be of lower order, cf. Assumption 3.3 below.' Assumption 3.3 indeed contains no lower-order or smallness condition on g1,g2; it only imposes Lipschitz/C^2 regularity. So the heuristic derivation of the critical space is not backed by the stated assumptions. Having checked the proof, however, this is not load-bearing. Theorems 3.4–3.5 are proved through the Agresti–Veraar fixed-point framework, whose own criticality condition (4.33) is verified in Section 4 for the noise terms: g1 uses ρ=0 and the same β as the deterministic nonlinearity; g2 (sv≤0) uses ρ=1 with equality; g2 (sv>0) uses subcritical parameters. These verifications hold for the most singular coefficients permitted by Assumption 3.3, e.g. g1(u,v,∇v)=∇v and g2(v)=v, because the estimates in Lemmas 4.4–4.6 are based on Sobolev embeddings that are exactly tuned to (3.7)–(3.10). Thus no noise term satisfying Assumption 3.3 is supercritical relative to the spaces (3.5). The gap is terminological: the paper should state that 'critical' is used in the AV sense (subcriticality condition (4.33)), not assert without proof that the noise is of lower order under the deterministic heat scaling.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the stochastic parabolic-parabolic Keller--Segel system (1.1) on the d-dimensional torus, d≥3, with nonlinear multiplicative noise. The main results are Theorem 3.4, which gives local L^p_κ-strong well-posedness in the scaling-critical Besov trace space B^{d/q-2}_{q,p}(T^d) × B^{d/q_v}_{q_v,p}(T^d) under the parameter conditions (3.3) and (3.6)--(3.10), and Theorem 3.5, which shows that sufficiently small initial data in this space imply, for any T<∞ and ε∈(0,1), that the maximal existence time σ satisfies P(σ≥T)≥1−ε. The proofs are carried out within the Agresti--Veraar stochastic maximal L^p_κ-regularity framework: Theorem 4.2 proves the required maximal regularity for the linear triangular operator, and Lemmas 4.3--4.6 establish the needed estimates for the deterministic chemotaxis nonlinearity and the two noise coefficients. The paper also contains a detailed parameter discussion (Remark 3.1) identifying the admissible dimensions and parameter regimes, including the uniqueness of the critical choice in d=3.","tokens_in":30876,"tokens_out":33973,"duration_ms":267137,"significance":"If correct, the results constitute a meaningful extension of deterministic critical-space Keller--Segel theory to the stochastic setting with a reasonably general class of nonlinear noise coefficients. The proof is systematic and unusually detailed for an Oberwolfach report contribution: the verification of the Agresti--Veraar hypotheses, especially the delicate subcriticality checks for the g2 noise term in the case s_v>0, is explicit and appears internally consistent. The small-data long-time existence result with high probability is also nontrivial. The main caveat is terminological: the 'critical spaces' are derived from the deterministic scaling, while the noise is only shown to be subcritical in the sense of the framework's condition (4.33). This should be clarified, but it does not affect the validity of the theorems.","major_comments":[{"comment":"The definition of L^p_κ-strong solution requires only g^1∈L^2((0,σ);H^{1+s,q}(T^d;ℓ_2)) and g^2∈L^2((0,σ);H^{1+s_v,q_v}(T^d;ℓ_2)). However, the proof of Theorem 3.4 and, in particular, the use of stochastic maximal regularity in Lemma 5.1 (see (5.26) and the subsequent application of Theorem 4.2) require the stronger L^p((0,σ),w_κ;γ(ℓ_2,X_{1/2}))-type regularity. The L^2 condition is not the one used in [AV22a, Definitions 4.3--4.4], and it is not clear that the stated maximality property in Definition 3.2(iii) is preserved if competitors are allowed to satisfy only the weaker L^2 condition. Please either align the definition with the framework's integrability requirement or explain why the weaker condition is sufficient for the stated uniqueness and maximality conclusions.","section":"Definition 3.2(i)"},{"comment":"The absorption in (5.17) is written with r_0, but the subsequent estimate on the event {τ≥T_0} uses the larger threshold \\tilde r_0, which was introduced just before (5.17). Since \\tilde r_0^p = 3/(4R) > 1/(2R) = r_0^p, the inequality (5.17) as stated does not directly apply to the argument with \\tilde r_0. This can be repaired by imposing (5.17) with \\tilde r_0 in place of r_0, or by choosing T_0 smaller by a constant factor. The gap is local and does not affect the main strategy, but the proof of Lemma 5.1 as written is incomplete at this point.","section":"Section 5.1, Eq. (5.17)"},{"comment":"The derivation of the critical spaces drops the noise terms with the sentence 'Neglecting contributions of the noise terms, which we take to be of lower order, cf. Assumption 3.3 below.' Assumption 3.3, however, contains no lower-order, smallness, or scaling condition on g^1 and g^2; it only imposes Lipschitz and C^2 regularity. This is not a correctness issue for Theorems 3.4--3.5, because Section 4 verifies the Agresti--Veraar (sub)criticality condition (4.33) for all noise coefficients admitted by Assumption 3.3. Nevertheless, the heuristic statement should be revised to say that the critical spaces are determined by the deterministic part and that the noise terms are verified to be subcritical with respect to condition (4.33) in Section 4. As written, the passage could be read as asserting an unproved scaling property of the noise.","section":"Section 3.1"}],"minor_comments":[{"comment":"The reference [ASV25] is described as a collaboration with 'the fourth author of the current manuscript'; Max Sauerbrey is the fifth author. Please correct the ordinal.","section":"Section 1.4"},{"comment":"The displayed identity before (4.28) should read H^{1+s_v,q_v}(T^d;ℓ_2) = [H^{1,q_v}(T^d;ℓ_2), H^{2,q_v}(T^d;ℓ_2)]_{s_v}, not with interpolation index s_v−1. The subsequent inequality (4.28) uses the correct powers 1−s_v and s_v, so this is a typographical error.","section":"Lemma 4.6, displayed interpolation"},{"comment":"In the definition of the stochastic integral, the lower limit should be a, not 0, to be consistent with the interval [a,b].","section":"Definition 4.1(i)"},{"comment":"The uniqueness assertion uses the same symbols (u,v,σ) for two solutions. Please use different notation, e.g. (u,v,σ) and (\\tilde u,\\tilde v,\\tilde σ), to avoid confusion.","section":"Theorem 3.4, uniqueness statement"},{"comment":"The phrase 'In the preceding Section 3' should read 'In the following Section 3'.","section":"Section 1.5"}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern about the 'lower order' noise assertion in §3.1 does not, on reading the proofs, translate into a correctness problem: Section 4 verifies the Agresti--Veraar subcriticality condition directly, and the estimates in Lemmas 4.4--4.6 cover the most singular coefficients permitted by Assumption 3.3. The issues I have identified are local and fixable. The manuscript would benefit from a careful pass to align the definition of solution with the framework's integrability spaces and to tighten the absorption argument in Lemma 5.1. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid paper. It proves local well-posedness for the stochastic parabolic–parabolic Keller–Segel system in d≥3 in the scaling-critical Besov trace space B^{d/q-2}_{q,p} × B^{d/q_v}_{q_v,p} with multiplicative noise, plus small-data long-time existence with high probability. The result is new: earlier stochastic KS papers were mostly 1D/2D. The proof is a careful application of Agresti–Veraar, and the hard part—estimating F and G in the right interpolation spaces—is done honestly. I checked the parameter bookkeeping and the embeddings; the arithmetic works.\n\nThe reader's main caveat is the sentence in §3.1 that says the noise is 'of lower order' and then derives the critical spaces by neglecting it. I don't think that is a real flaw. The derivation is heuristic, but the proof never relies on it. Section 4 verifies the AV criticality condition (4.33) directly for the worst-case coefficients satisfying Assumption 3.3: g1 with ρ=0, g2 with ρ=1 in the s_v≤0 case, and subcritical parameters in the s_v>0 case. So no admissible noise term is supercritical. The only fix needed is to rephrase that sentence as 'subcritical in the AV sense, see Section 4.'\n\nMinor issues: the introduction calls [ASV25] a collaboration with 'the fourth author'—Sauerbrey is the fifth. And the flagged exponent in (5.17) is not a typo; p(1+s_v) is the right exponent when you absorb Z^{2+s_v} into Z using Z ≤ r_0^p. I'd leave it alone.\n\nBottom line: this is a solid contribution for the SPDE/critical-spaces community. It doesn't introduce a new mechanism or solve a major open problem, but it fills a clear gap and does so cleanly. The central argument holds up. I'd send it to a serious referee, and I'd put it on a reading-group list for people who want to see a worked AV application.","headline":"Solid, careful AV-application that fills the d≥3 stochastic Keller–Segel critical-space gap; the flagged noise-subcriticality concern is cosmetic, not load-bearing.","tokens_in":31438,"tokens_out":9005,"would_cite":true,"duration_ms":70734,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","35R60","35A01","92C17","35Q92"],"pacs":[],"model":"deepseek-v4-flash","headline":"The stochastic Keller–Segel system is locally well-posed in scaling-critical Besov spaces on the d-dimensional torus for d≥3, and small critical data survive arbitrarily long with high probability.","keywords":["Keller–Segel equation","environmental noise","well-posedness","stochastic maximal regularity","critical Besov spaces","small data","chemotaxis","torus"],"falsifier":"Compute the scaling homogeneity of the noise terms for g1(u,v,∇v)=u∇v and g2(v)=v under the rescaling u→λu, v→v, x→λ^{1/2}x, t→λt: both noise components scale like the deterministic terms, so the noise is critical, not lower order. If the theory of Section 4 is correct, local well-posedness must fail for this (non-Lipschitz) example unless additional regularity is assumed, revealing the subcriticality assumption as essential. Alternatively, search for a Lipschitz pair (g1,g2) satisfying Assumption 3.3 and (3.19) that reaches critical scaling; a positive construction would directly falsify the","tokens_in":30400,"feed_emoji":"🦠","tokens_out":13569,"duration_ms":124451,"temperature":0.7,"pith_summary":"The paper studies the stochastic parabolic-parabolic Keller–Segel equations on the d-dimensional torus for d≥3, where the cell density u obeys a heat equation with chemotactic drift −div(u∇v) and a Lipschitz noise, while the chemoattractant v obeys a damped heat equation with another noise. Treating the noise as a lower-order perturbation, the authors identify the scaling-critical Besov spaces for the initial data, B^{d/q−2}_{q,p}(T^d) × B^{d/q_v}_{q_v,p}(T^d), and prove that every initial datum in these spaces generates a unique maximal L^p_κ-strong solution — local well-posedness. They further show that small data in those critical spaces produce arbitrarily long lifetimes with probability arbitrarily close to one. This matters because it transplants the deterministic critical-space well-posedness theory of chemotaxis to a stochastic setting and gives a quantitative probabilistic handle on the maximal existence time.","feed_headline":"Stochastic Keller–Segel tamed in critical spaces","feed_subtitle":"Solutions exist in critical Besov spaces; small data live long with high probability.","key_machinery":"The load-bearing object is the scaling-critical trace space X_{1−(1+κ)/p,p} = B^{s+2−2(1+κ)/p}_{q,p}(T^d) × B^{s_v+2−2(1+κ)/p}_{q_v,p}(T^d), which under conditions (3.7)–(3.8) coincides with the scaling-invariant Besov space B^{d/q−2}_{q,p}(T^d) × B^{d/q_v}_{q_v,p}(T^d). The machinery has three parts: (i) the triangular structure of −A, which lets the proof solve the u-equation first and insert it into the v-equation via an Itô transformation; (ii) sharp Sobolev embeddings and product estimates for the nonlinearity and the noise, with parameter choices that make the subcriticality condition hold with equality for the deterministic part; and (iii) an interval-iteration argument that concatena","core_discovery":"The central discovery is that the stochastic Keller–Segel system admits an L^p_κ-maximal solution with initial data in the scaling-critical trace space B^{d/q−2}_{q,p}(T^d) × B^{d/q_v}_{q_v,p}(T^d) for d≥3, under the parameter conditions (3.3) and (3.6)–(3.10) and Lipschitz noise coefficients (with g2 one derivative smoother). The proof shows the linear block operator −A = [[Δ,0],[I,Δ−I]] has stochastic maximal L^p_κ-regularity via its triangular structure, and that the deterministic nonlinearity div(u∇v) and the noise terms satisfy the sharp product and growth estimates at the critical index. The companion small-data result states that if the noise coefficients vanish at zero, then for ever","pith_inferences":["A natural extension is to classify which noise coefficients g1,g2 are subcritical under the heat scaling, since the current argument assumes (rather than proves) this 'lower order' property, and to examine behavior at the critical threshold.","The triangular linear structure is likely to extend the same argument to other coupled parabolic systems with lower-triangular linear parts, such as chemotaxis-fluid or reaction-diffusion systems, wherever the same trace-space framework applies.","The interval-iteration in the small-data proof could be refined to yield explicit lower bounds on P(σ≥T) in terms of the critical norm and the noise coefficients, which would be useful for numerical simulation of blow-up.","The parameter incompatibility in d=2 suggests a separate treatment with logarithmically critical spaces, paralleling the deterministic Moser–Trudinger-based theory; the current result leaves that open."],"forward_implications":["For d=3 the theorem uniquely identifies the critical space H^{−1/2,2}(T^3) × H^{3/2,2}(T^3) with p=2, κ=0, so stochastic Keller–Segel well-posedness results in three dimensions must be formulated in this space.","Small critical data yield existence up to any prescribed T with probability arbitrarily close to 1, together with a stopped L^p estimate — a quantitative, probabilistic control on the maximal existence time rather than only a qualitative local statement.","The results extend the deterministic Keller–Segel critical-space theory, showing that subcritical Lipschitz environmental noise does not destroy the scaling structure of the equation.","For d≥4 the parameter conditions permit a range of Besov spaces, giving flexibility in the functional setting for applications.","The solution is unique, maximal, and depends continuously on the initial data within the class of L^p_κ-local solutions."],"fun_headline_variants":["Stochastic Keller–Segel: critical-space existence","Critical Besov well-posedness for stochastic Keller–Segel","Small-data stochastic Keller–Segel: long life, high probability","Noise-tolerant Keller–Segel: critical space proof","Stochastic Keller–Segel: local solutions in critical spaces"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The noise terms are assumed to be of lower order than the deterministic part under the heat scaling, so that the critical spaces are determined by the deterministic equation; this subcriticality is asserted in Section 3.1 rather than proved from the structure of g1 and g2, and if the noise were critical or supercritical the trace space (3.5) would not be the correct critical space.","fun_headline_variants_meta":{"raw":{"variants":["Stochastic Keller–Segel: critical-space existence","Critical Besov well-posedness for stochastic Keller–Segel","Small-data stochastic Keller–Segel: long life, high probability","Noise-tolerant Keller–Segel: critical space proof","Stochastic Keller–Segel: local solutions in critical spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000981,"raw_usage":{"total_tokens":3958,"prompt_tokens":660,"completion_tokens":3298,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":404,"completion_tokens_details":{"reasoning_tokens":3213}},"tokens_in":404,"tokens_out":3298,"duration_ms":25009,"temperature":1.0,"reasoning_tokens":3213,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:21:39.401994+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the scaling homogeneity of the noise terms for g1(u,v,∇v)=u∇v and g2(v)=v under the rescaling u→λu, v→v, x→λ^{1/2}x, t→λt: both noise components scale like the deterministic terms, so the noise is critical, not lower order. If the theory of Section 4 is correct, local well-posedness must fail for this (non-Lipschitz) example unless additional regularity is assumed, revealing the subcriticality assumption as essential. Alternatively, search for a Lipschitz pair (g1,g2) satisfying Assumption 3.3 and (3.19) that reaches critical scaling; a positive construction would directly falsify the","supporting_citations":[],"review_version":1}