{"id":"31c81b4a-1039-4070-8c8d-7133b3897c10","arxiv_id":"2508.05032","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors identify exact local and uniform spatio-temporal moduli of continuity for nonlinear parabolic SPDEs on bounded intervals and for the open KPZ equation, using new strong local non-determinism proofs under Robin boundary conditions.","lead":"This paper proves exact rates at which solutions to nonlinear stochastic heat equations wiggle in space and time, on a bounded interval with Dirichlet, Neumann, or Robin boundary conditions. The same rates are shown for the open KPZ equation, a model of interfaces with boundary driving.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.10 is stated for the full rectangle [0,T]×[0,L] but its proof relies on Proposition 4.9, which only applies to [a,T]×[0,L] (a>0) or [0,T]×[c,d] under (1.6); the gap affects the transfer of moduli at t=0/boundaries.","rationale":"The reader's weakest_assumption focused on the polarity condition σ^{-1}{0} polar for u. That condition is explicit in Theorem 1.2 and is a genuine limitation, but it is not a hidden assumption or a proof gap: the theorem is conditional, and the paper gives examples where it holds. The more load-bearing issue is the proof of Proposition 4.10, which is the key step that transfers exact moduli from the Gaussian SHE to the nonlinear SPDE. The statement of Proposition 4.10 covers [0,T]×[0,L] without any initial-data condition, but its only tool, Proposition 4.9, is proved only for strictly positive times (or for the t=0 case under (1.6)). Thus the proof of Proposition 4.10 has a real gap exactly in the regime where the deterministic initial data can be non-Lipschitz (t=0) and where the spatial boundary can enter. The main theorems for interior times (t0>0, a>0) avoid the gap by localizing to [a,T]×[c,d], so the central claim (Theorem 1.1 for z0∈(0,∞)×(0,L)) is unaffected. The t=0 extensions in the paper explicitly impose (1.4),(1.6),(1.10), which would allow a local version of Proposition 4.10; but the lemma as stated and proved does not reflect that. For these reasons, the verdict should remain ACCEPT (UNCHANGED) rather than CONDITIONAL, since the central result for interior points is solid and the gap is patchable in the intended applications. I also noted a sign typo in the displayed identity in the proof of Proposition 4.9 and a false 'single d-ball' covering claim in the t=0 case of Theorem 3.14, but neither affects the final estimates.","tokens_in":46616,"tokens_out":38695,"duration_ms":387493,"concrete_test":"Rewrite the proof of Proposition 4.10 by covering [0,T]×[0,L] with countably many admissible rectangles: e.g., [1/n,T]×[1/n,L−1/n] for n≥1, together with [0,T]×[c_k,d_k] where (1.6) holds, and verify that the Borel–Cantelli bound of Proposition 4.9 can be applied on each piece and summed over the countable cover. If the sum converges (or the theorem is only applied to a fixed interior rectangle), the gap is cosmetic; if not, construct a counterexample to Proposition 4.10 with u0∈L² that violates (1.6) at t=0 and show the deterministic increment G_t*u0 dominates the claimed o(ρ^p) rate, invalidating the lemma as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central transfer from the Gaussian field w to the nonlinear solution u uses Proposition 4.10, which asserts that the linearization error E(z;z') is o(ρ^p) a.s. uniformly on [0,T]×[0,L] for any p<ζ, with no additional condition on u0. The proof of Proposition 4.10 invokes Proposition 4.9, but Proposition 4.9 is explicitly stated only for two types of regions: I=[a,T]×[0,L] with a>0, or I=[0,T]×[c,d] (c<d) under assumption (1.6). Neither covers the full rectangle [0,T]×[0,L]. The truncation argument in Proposition 4.10 replaces b,σ by bounded truncations but does not introduce (1.6), so the Borel–Cantelli step is unjustified on the parts of the rectangle with t=0 or at the spatial boundary. In the main theorems for interior points (t0>0, x0∈(0,L)), this is harmless because the proof localizes to a subrectangle [a,T]×[c,d] where Proposition 4.9 applies. For the t0=0 extensions, the paper adds (1.4)/(1.6)/(1.10), which would restore locality. However, as stated, Proposition 4.10 overclaims, and the proof as written does not establish the lemma on the full domain. This is the weakest link in the chain from Gaussian moduli to nonlinear moduli, because Proposition 4.10 is the vehicle that makes the linearization error negligible at the exact rate.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies exact spatio-temporal moduli of continuity for solutions to nonlinear parabolic SPDEs on a bounded interval under Dirichlet, Neumann, or Robin boundary conditions. The main results are: a Khinchine-type local law of the iterated logarithm (Theorem 1.1), an exact uniform modulus of continuity (Theorem 1.2), existence and density of exceptional increments (Corollary 1.4), small-ball probability estimates (Theorem 1.7), and a Chung-type LIL (Theorem 1.8). These results are extended to the open KPZ equation via the Hopf-Cole transform (Theorems 1.11 and 1.13). The proof strategy is: (i) establish strong local non-determinism and matching variance bounds for the linear stochastic heat equation, (ii) verify the Lee--Xiao framework for anisotropic Gaussian random fields, (iii) prove detailed linearization error estimates with exponent ζ>1, and (iv) transfer exact moduli from the Gaussian solution w to the nonlinear solution u. The paper is well structured and contains substantial new technical work, especially the SLND proof under Robin boundary conditions.","tokens_in":47033,"tokens_out":8210,"duration_ms":96333,"significance":"If the proofs are completed as claimed, this is a substantial advance: it gives the first exact spatio-temporal moduli of continuity for non-Gaussian parabolic SPDEs on bounded intervals and for the open KPZ equation, with constants that are universal up to the law of the driving noise. The paper is notable for being parameter-free: no fitted constants or circular definitions appear, and the Gaussian component is imported from an external general framework (Lee--Xiao) whose hypotheses are verified rather than modified. The new SLND result for Robin boundary conditions, obtained through the eigenfunction basis rather than Fourier transform, is an original contribution that may be useful beyond this paper. The linearization-error estimates, with explicit exponents giving ζ>1, are the key technical engine and appear sound in the interior case. The main obstruction is a proof gap in the full-domain version of the linearization-error almost-sure bound, which affects some boundary and near-t=0 statements as written.","major_comments":[{"comment":"Proposition 4.10 is stated for the full rectangle [0,T]×[0,L], but its proof invokes Proposition 4.9 with I=[0,T]×[0,L]. Proposition 4.9 is explicitly proved only for I=[a,T]×[0,L] with a>0, or for I=[0,T]×[c,d] under assumption (1.6). The truncation argument for general b,σ does not introduce (1.6), so the Borel–Cantelli step is not justified near t=0 or at the spatial boundary. This is load-bearing because Proposition 4.10 is the vehicle used to make the linearization error negligible at the exact logarithmic rates. The interior results (t0>0, x0∈(0,L)) can be recovered by localizing to a subrectangle [a,T]×[c,d] where Proposition 4.9 applies, but the stated full-domain proposition and the a=0 extensions are not proved as written. The authors should either prove the full-domain statement with a genuinely new argument, or restate and use a local version that is actually covered by Propo","section":"§4.2, Proposition 4.10"},{"comment":"The proof of Theorem 1.1 for t0=0 says that Proposition 4.10 and Theorem 3.14 'continue to hold when t0=0' under assumption (1.4). However, the only route to Proposition 4.9 on a rectangle touching t=0 requires assumption (1.6) globally on [0,T]×[c,d], whereas (1.4) is a local one-point condition near (0,x0). It is not shown that (1.4) implies the global condition needed for the error estimate. This is a mismatch between hypothesis and proof. A local version of the error estimate should be stated and proved, or assumption (1.6) should be imposed. The same issue affects the t0=0 statements of Theorems 1.7 and 1.8, where the displayed tail estimate for E(z0;z) relies on Proposition 4.9 in a form not justified under (1.10).","section":"§5.1, Theorem 1.1 (t0=0 case)"},{"comment":"The uniform-modulus theorem for a=0 requires the full-domain linearization error to be o(ρ sqrt(log(1/ρ))) on [0,T]×[c,d]. The proof invokes Proposition 4.10, which has the gap described above. The additional assumption (1.6) does supply the hypothesis needed by Proposition 4.9 on I=[0,T]×[c,d], so the a=0 case may be fixable by a direct appeal to Proposition 4.9 rather than Proposition 4.10. As written, however, Proposition 4.10 overclaims, and the proof of Theorem 1.2 does not clearly isolate the range of ε and the rectangle where the error estimate is valid. The authors should make the local/global distinction explicit and ensure the cited proposition exactly matches the domain on which it is applied.","section":"§5.2, Theorem 1.2 (a=0 case)"}],"minor_comments":[{"comment":"The proof refers to 'Theorem 1.3' when showing that the fixed-point limsup with denominator ρ sqrt(log(1/ρ)) is zero a.s. There is no Theorem 1.3; the intended reference is almost certainly Theorem 1.1 combined with the observation that sqrt(log log(1/ρ))/sqrt(log(1/ρ))→0. Please correct the reference.","section":"§5.3, Corollary 1.4 proof"},{"comment":"The symbol c is overloaded: it is used both for the spatial lower bound in intervals like [0,T]×[c,d] and for the localization parameter in the proof. This makes the proof harder to follow. Rename one of them.","section":"§4.1, Lemma 4.7"},{"comment":"The statement says 'for any 0<a<T' but the case I=[0,T]×[c,d] under (1.6) is also included. The phrase 'This remains valid when I=[0,T]×[c,d]...' is clear, but it would be useful to state explicitly that the bound is uniform in t∈[a,T] in the first case and t∈[0,T] in the second. Currently the parameter a in the first case is not used in the local estimates except through Lemma 2.3.","section":"§4.1, Lemma 4.6"},{"comment":"In the K0>0 part, the argument considers {w(t,x0)} and applies a one-dimensional LIL. It would be helpful to state explicitly that the constant K0 in (3.20) is bounded below by the one-dimensional constant, since the sup over the full parabolic ball dominates the temporal sup. This is implicit but should be spelled out.","section":"§3.4, Theorem 3.14 proof"},{"comment":"The proof writes 'Let A denote the event appearing on the left-hand side of (4.11)' and then defines B0. In the displayed bound for P{B0}, the exponent of δ in #J is fine, but the notation for the exponential term would be clearer if the same h were used consistently with the statement of the proposition. Currently the reader must reconcile h and δ.","section":"§4.2, Proposition 4.9 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is strong and the interior results are very likely correct, but the gap in Proposition 4.10 is real and is used in several statements. I recommend major revision rather than rejection because the issue appears fixable by restricting the claims to domains covered by Proposition 4.9, or by adding a genuinely local lemma for the linearization error. The authors should also carefully re-check the t0=0 extensions: (1.4) and (1.10) are local conditions and do not obviously imply the global condition (1.6) used in Proposition 4.9. There is no sign of circularity or fitted parameters; the main technical architecture is sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real theorem paper. The authors prove exact spatio-temporal moduli of continuity, LILs, small-ball estimates, and Chung-type LILs for nonlinear parabolic SPDEs on bounded intervals with Dirichlet, Neumann, and Robin conditions, and they push most of this through to the open KPZ equation via Hopf-Cole. The results are genuinely new: prior work mostly handled R or T, and the Robin case has no Fourier transform available, so the new eigenfunction-based SLND proof is a legit technical contribution. The transfer from the Gaussian field to the nonlinear solution through linearization error estimates with Holder exponent >1 is well organized. The Gaussian machinery comes from Lee-Xiao, and citing it is appropriate; they verify the hypotheses carefully rather than just importing conclusions.\n\nThe main soft spot is Proposition 4.10. As stated, it claims a.s. uniform vanishing of the linearization error on the full rectangle [0,T]x[0,L]. The proof reduces to Proposition 4.9, which is only proved for [a,T]x[0,L] with a>0 or [0,T]x[c,d] under the extra condition (1.6). The truncation argument in Prop 4.10 doesn't add (1.6), so the Borel-Cantelli step isn't justified at t=0 or at the spatial boundary. For the interior theorems (t0>0, x0 in (0,L)), this is harmless—the proof only needs the error on a small ball away from t=0, and you can get that from Prop 4.9 directly. For the t=0 extensions, the paper's added assumptions (1.4)/(1.10) look designed to restore (1.6) locally, but the text doesn't say that, and as written Prop 4.10 overreaches. This is a fixable scope error, not a load-bearing flaw.\n\nOne other thing: the uniform-modulus theorem requires sigma^{-1}{0} polar for u, and the paper notes this. That's explicit and fine, though for multiplicative noise it's checked only for positive data.\n\nIf I were the editor, I'd send it to a serious referee. The main results are significant and the architecture is sound; the referee should ask for a corrected Prop 4.10 with a clear domain statement and a proof that matches the local statements actually used.","headline":"Solid paper with new exact moduli for nonlinear SPDEs on bounded intervals; the main results look right, but Proposition 4.10 overclaims domain scope and needs a patch for boundary extensions.","tokens_in":47481,"tokens_out":4561,"would_cite":true,"duration_ms":47467,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60G17"],"pacs":[],"model":"deepseek-v4-flash","headline":"For nonlinear parabolic SPDEs on a bounded interval, the paper identifies exact local and uniform spatio-temporal moduli of continuity matching the linear Gaussian field, and carries them to the open KPZ equation.","keywords":["stochastic partial differential equation","stochastic heat equation","open KPZ equation","law of the iterated logarithm","modulus of continuity","small-ball probabilities","strong local non-determinism","Robin boundary conditions"],"falsifier":"Take $\\sigma(u)=u$, $b=0$, zero initial data $u_0=0$, and Neumann boundary: the unique solution is $u\\equiv0$, so $\\sigma^{-1}\\{0\\}$ is not polar and $|\\sigma(u(z))|=0$ everywhere; the quotient in (1.5) is not defined at any point. This example shows the polarity hypothesis is genuinely needed, and any claim that (1.5) holds without it can be refuted by checking this case.","tokens_in":46531,"feed_emoji":"📈","tokens_out":8015,"duration_ms":80774,"temperature":0.7,"pith_summary":"The paper tries to establish that the sample functions of a broad class of nonlinear parabolic SPDEs on a bounded interval oscillate at exactly the rates of the underlying linear stochastic heat equation, with only the noise coefficient $\\sigma$ evaluated at the solution as a modulating factor. At a fixed space-time point, spatio-temporal increments are almost surely of size $|\\sigma(u(z_0))|\\,\\rho(z,z_0)\\sqrt{\\log\\log(1/\\rho(z,z_0))}$; uniformly over an interior rectangle, the increments are of size $|\\sigma(u(z))|\\,\\rho(z,z')\\sqrt{\\log(1/\\rho(z,z'))}$ with a positive finite constant. These rates are sharp, so the nonlinearity and drift do not change the leading oscillation law. The results imply the existence of random exceptional points where increments are anomalously large, and prove matching small-ball probabilities and a Chung-type law of the iterated logarithm. The same moduli, with the same constants, hold for the open KPZ equation through its Hopf-Cole solution.","feed_headline":"Parabolic SPDEs get exact spatio-temporal oscillation rates","feed_subtitle":"Nonlinear heat SPDEs oscillate at linear Gaussian rates; open KPZ inherits the exact moduli.","key_machinery":"Strong local non-determinism (SLND): the property that the conditional variance of a Gaussian field at a point given finitely many other points is bounded below by a constant times the minimum of the increment variances. The paper proves sharp SLND for the linear stochastic heat equation under Dirichlet, Neumann, and Robin boundary conditions, with matching two-point variance bounds $\\mathrm{Var}(w(t,x)-w(s,y))\\asymp \\rho^2((t,x),(s,y))$ up to boundary factors. This is combined with detailed linearization-error estimates showing that $E(z;z'):=u(z')-u(z)-(G*u_0)(z')+(G*u_0)(z)-\\sigma(u(z))(w(z')-w(z))$ is $o(\\rho^p)$ for every $p<\\zeta$ with $\\zeta>1$, almost surely. These two ingredients le","core_discovery":"Let $u$ be the mild solution to $\\partial_t u=\\frac12\\partial_x^2u+b(u)+\\sigma(u)\\xi$ on $(0,L)$ under Dirichlet, Neumann, or Robin boundary conditions. The paper's main claim is that for every fixed $z_0=(t_0,x_0)\\in(0,\\infty)\\times(0,L)$ there is a constant $K_0\\in(0,\\infty)$ such that $$\\lim_{\\varepsilon\\to0+}\\sup_{z\\in B^*_\\rho(z_0,\\varepsilon)}\\frac{|u(z)-u(z_0)|}{\\rho(z,z_0)\\sqrt{\\log\\log(1/\\rho(z,z_0))}}=K_0|\\$\\sigma$(u(z_0))|\\quad\\text{a.s.}$$ and, under the assumption that $\\sigma^{-1}\\{0\\}$ is polar for $u$, for every interior rectangle $I$ there is $K\\in(0,\\infty)$ such that $$\\lim_{\\varepsilon\\to0+}\\sup_{z,z'\\in I:0<\\rho(z,z')\\le\\varepsilon}\\frac{|u(z')-u(z)|}{|\\$\\sigma$(u(z))|\\rho(z,z","pith_inferences":["The method's reliance on crude heat-kernel bounds rather than Gaussian bounds suggests the same exact-moduli transfer should work for more general second-order operators and rough domains, provided an SLND estimate can be proven; this is an extension the paper itself leaves implicit.","The constants $K_0$ and $K$ are proven finite and positive but not computed; an editorial guess is that $K$ is the parabolic-metric analog of the Brownian modulus constant, so it may be expressible in terms of metric entropy of the rectangle.","For a general Lipschitz $\\sigma$ that attains zero, the uniform modulus with denominator $|\\sigma(u(z))|$ cannot be the right normalization: points where $\\sigma(u(z))=0$ would make the ratio blow up or be undefined, so a modified statement with a different weight would be needed.","The small-ball exponent $1/6$ reflects the parabolic scaling dimension $1+2=3$ in the exponent $(r/\\varepsilon)^6$; this suggests that for colored noise or different boundary geometries the exponents change by the effective dimension of the driving noise."],"forward_implications":["Sample paths are in $\\bigcap_{\\alpha<1/4,\\beta<1/2}C^{\\alpha,\\beta}(I)$ but not in $C^{1/4,1/2}(I)$; the logarithmic correction is sharp, not an artifact of the proof.","There exist random, dense, Lebesgue-null exceptional sets inside every interior rectangle at which spatio-temporal increments exceed the fixed-point law of the iterated logarithm by a logarithmic factor.","Matching small-ball bounds hold with exponent $(r/\\varepsilon)^6$, giving a Chung-type LIL: $\\liminf_{\\varepsilon\\to0+}(\\log\\log(1/\\varepsilon))^{1/6}\\varepsilon^{-1}\\sup_{B_\\rho(z_0,\\varepsilon)}|u-u(z_0)|=C_2|\\sigma(u(z_0))|$ a.s.","For the open KPZ equation, the same local and uniform moduli and Chung-type LIL hold with the same constants as for the linear stochastic heat equation.","The full statements are valid under Dirichlet, Neumann, and Robin boundary conditions; under Dirichlet and Neumann the variance and SLND bounds match up to the boundary and to $t=0$."],"supporting_citations":[{"why":"Supplies the exact local and uniform moduli of continuity, small-ball constants, and Chung-type LIL for anisotropic Gaussian random fields that are applied to the linear stochastic heat equation.","marker":"[53]"},{"why":"Defines the open KPZ equation, sets the Robin-boundary stochastic heat equation formulation, and provides the strict positivity of the Hopf-Cole solution used in the KPZ transfer.","marker":"[20]"},{"why":"Provides the linearization and heat-kernel localization method for nonlinear SPDEs that is adapted here to spatio-temporal increments on bounded intervals.","marker":"[31]"},{"why":"Supplies the Burkholder-Davis-Gundy inequality and moment estimates for stochastic integrals used throughout the linearization-error estimates.","marker":"[42]"},{"why":"Gives the Walsh theory of space-time white noise, mild solutions, and stochastic integration that defines the solution and the linearization error.","marker":"[70]"},{"why":"Contributes the Orey-Taylor exceptional-set argument that is adapted to prove the density of exceptional spatio-temporal points in Corollary 1.4.","marker":"[64]"}],"fun_headline_variants":["Exact continuity moduli for nonlinear heat SPDEs","Open KPZ inherits exact oscillation behavior","SPDE increments follow precise log-log laws","Uniform oscillation rates for SPDE solutions","Random spacetime extremes in SPDE sample paths"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The uniform-modulus and exceptional-set results require that the set where $\\sigma$ vanishes is polar for $u$, so $|\\sigma(u(z))|$ stays almost surely bounded away from zero on each interior interval; for a general Lipschitz $\\sigma$ this is not automatic, and without it the right side of (1.5) is undefined.","fun_headline_variants_meta":{"raw":{"variants":["Exact continuity moduli for nonlinear heat SPDEs","Open KPZ inherits exact oscillation behavior","SPDE increments follow precise log-log laws","Uniform oscillation rates for SPDE solutions","Random spacetime extremes in SPDE sample paths"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00031,"raw_usage":{"total_tokens":1625,"prompt_tokens":785,"completion_tokens":840,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":772}},"tokens_in":529,"tokens_out":840,"duration_ms":9092,"temperature":1.0,"reasoning_tokens":772,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T23:36:36.061448+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $\\sigma(u)=u$, $b=0$, zero initial data $u_0=0$, and Neumann boundary: the unique solution is $u\\equiv0$, so $\\sigma^{-1}\\{0\\}$ is not polar and $|\\sigma(u(z))|=0$ everywhere; the quotient in (1.5) is not defined at any point. This example shows the polarity hypothesis is genuinely needed, and any claim that (1.5) holds without it can be refuted by checking this case.","supporting_citations":[{"cited_title":"1, 523–550","cited_arxiv_id":null,"evidence_quote":"Supplies the exact local and uniform moduli of continuity, small-ball constants, and Chung-type LIL for anisotropic Gaussian random fields that are applied to the linear stochastic heat equation."},{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"Defines the open KPZ equation, sets the Robin-boundary stochastic heat equation formulation, and provides the strict positivity of the Hopf-Cole solution used in the KPZ transfer."},{"cited_title":"1180, Springer, Berlin, 1986, pp","cited_arxiv_id":null,"evidence_quote":"Gives the Walsh theory of space-time white noise, mild solutions, and stochastic integration that defines the solution and the linearization error."},{"cited_title":"James Taylor, How often on a Brownian path does the law of iterated logarithm fail?, Proc","cited_arxiv_id":null,"evidence_quote":"Contributes the Orey-Taylor exceptional-set argument that is adapted to prove the density of exceptional spatio-temporal points in Corollary 1.4."}],"review_version":2}