{"id":"a9036906-d699-4a71-8530-e8c92434486f","arxiv_id":"2508.11859","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A sharp upper bound on hitting probabilities for nonlinear stochastic heat equations is proved, completing the sharp lower bound from the authors' earlier work.","lead":"This math paper proves a sharp upper bound on the probability that a solution to a nonlinear stochastic heat equation hits a fixed set, matching an earlier sharp lower bound. It uses Malliavin calculus to estimate a joint density involving the solution and the supremum of a related linear equation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sharpness of the upper bound hinges on unverified Malliavin estimates for the supremum component; test against the Gaussian (zero nonlinearity) limit.","rationale":"The reader's weakest_assumption is exactly the load-bearing premise: the local nondegeneracy of the 2D vector and the finiteness/scaling of the Malliavin-estimate terms. My review agrees with this assessment. Since the full text is not available, I cannot identify a more specific internal flaw; the abstract gives only the strategy, not the inequalities. The most useful check is to force the method to reproduce a known result in the Gaussian limit, where the nonlinearity vanishes and the vector takes a simpler but still nontrivial form. This directly tests the same estimates that are needed for the nonlinear case, and it does not depend on unpublished details beyond the method already described in the abstract. The verdict remains UNVERDICTED: the central claim is plausible and supported by the authors' expertise, but the critical technical estimates are not inspectable from the abstract alone. No independent objection beyond the reader's is raised; the concern is a verification gap rather than a demonstrated error.","tokens_in":1027,"tokens_out":3804,"duration_ms":51881,"concrete_test":"Specialize to the case where the nonlinearity coefficient is zero, so that the 'nonlinear' solution u coincides with the linear solution v. Then the vector becomes (v(0,0), sup_R v) for a small rectangle R of side ℓ. For this Gaussian setting, classical sharp upper bounds on the joint density and on hitting probabilities are known. Apply the paper's Malliavin density formula and its estimates verbatim to this degenerate nonlinearity case—without using any nonlinear-specific simplifications—and compare the resulting bound on the joint density (or hitting probability) with the known Gaussian sharp bound as ℓ→0. If the paper's bound recovers the correct exponent and constant (up to universal constants), the concern is resolved. If it gives a strictly worse exponent, or if any of the intermediate terms fails to be finite unless additional non-smoothness assumptions are imposed, the sharpness","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—a sharp upper bound on hitting probabilities for nonlinear SHE—depends on a bound for the joint density of the 2D vector (u(t,x), sup_R v), where u is the nonlinear solution and v is the linear solution. The abstract states that this bound comes from a formula expressing the density of a locally nondegenerate random vector as an iterated Skorohod integral, and that the 'main effort' is estimating each term from that formula. Two load-bearing assumptions are implicit: (i) the vector (u(t,x), sup_R v) is locally nondegenerate in the sense required for the formula, and (ii) every term produced by the formula is finite and scales with the size ℓ of the rectangle R as ℓ^{-κ} with the sharp exponent κ taken from Gaussian theory. Since the full text is unavailable, neither can be verified. In particular, the second component is a supremum over a rectangle—a non-smooth functional that lacks Malliavin differentiability in the usual sense. If the formula requires both components to be smooth, a smoothing/regularization argument is needed, and the estimates must survive the limiting procedure with the correct scaling. If any term diverges or produces an extra logarithmic factor or a non-sharp power of ℓ, the upper bound is not sharp and the main claim fails even though the argument might yield a finite but non-sharp bound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper (arXiv:2508.11859, abstract-only) claims a sharp upper bound on hitting probabilities for solutions to nonlinear systems of stochastic heat equations on the line, complementing a sharp lower bound obtained earlier by two of the authors. The argument is stated to proceed through a bound on the joint density of a two-dimensional random vector whose components are the nonlinear SHE solution at a point and the supremum over a small rectangle of the linear SHE solution. The density bound is obtained from an iterated Skorohod integral formula for locally nondegenerate random vectors, and the main effort is described as estimating each resulting term via Malliavin calculus. Since only the abstract is available, the actual estimates, the nondegeneracy conditions, and the scaling of the bound with rectangle size cannot be inspected.","tokens_in":1150,"tokens_out":2354,"duration_ms":33159,"significance":"If the claimed result holds, it would close the gap between Gaussian and non-Gaussian hitting probability bounds for a broad class of nonlinear SPDEs, giving two-sided bounds with sharp dependence on the spatial scale. The proof strategy is credible: the density formula and Malliavin calculus are standard tools, and the authors have established relevant lower bounds previously. The abstract promises a parameter-free, sharp result, which is a valuable contribution. However, the technical core—the Malliavin estimates for the supremum component and the verification that the density formula applies to this non-smooth functional—is not visible from the abstract, so the significance is conditional on those estimates being correct.","major_comments":[{"comment":"The abstract states that the density bound applies to a random vector whose second component is the supremum of the linear SHE over a rectangle. A supremum is not Malliavin differentiable in the usual sense, so the iterated Skorohod integral formula cannot be applied directly unless some regularization is introduced. The abstract does not describe this regularization, nor does it state that the estimates survive the limiting procedure uniformly and with the correct scaling. This is load-bearing: if the regularization introduces an extra factor or if the estimates are not uniform, the sharp exponent could be lost.","section":"Abstract"},{"comment":"The claimed 'sharp' upper bound must match the lower bound's dependence on the rectangle size, typically a power of the side length. The abstract only says the density is bounded 'in terms of the size of the rectangle' and does not state the exponent, the constants, or the admissible class of nonlinearities/initial conditions. Without this precise statement, the sharpness claim cannot be verified. In particular, the zero-nonlinearity limit (u=v) should reproduce the known Gaussian supremum density bound; the manuscript should include this check explicitly.","section":"Abstract"},{"comment":"The abstract refers to 'nonlinear systems of stochastic heat equations' but the density bound is described for a two-dimensional random vector with one nonlinear solution and one linear supremum. If the intended application is to systems, the random vector would need to have dimension at least 2d, and the nondegeneracy conditions would involve the covariance structure of the system. The abstract should clarify whether the proof is for a single equation or a true system, or explain how the two-dimensional bound suffices in the system setting. This is a clarity issue that affects the reader's ability to evaluate the scope of the result.","section":"Abstract"}],"minor_comments":[{"comment":"The term 'locally nondegenerate' is used without definition; a brief description of the nondegeneracy condition (e.g., Malliavin matrix invertibility and integrability bounds) would make the abstract self-contained.","section":"Abstract"},{"comment":"The phrase 'a previous paper by two of the authors' should include a citation or reference, as the lower bound is a key benchmark for sharpness.","section":"Abstract"},{"comment":"The abstract does not state the regularity assumptions on the nonlinearity and the initial data; these are likely needed for the Malliavin estimates and should be mentioned.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only review; the full text was not available. The described program is plausible and no error is apparent from the abstract, but the central technical claims (the applicability of the density formula to the supremum functional, the finiteness of all Malliavin terms, and the sharp scaling) cannot be checked. I recommend obtaining the full manuscript before a substantive decision. If the full proof delivers the claimed uniform estimates and the Gaussian-limit check, the paper would likely be a strong accept."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You asked what I make of arXiv:2508.11859. Short version: if the proof works, this is a genuine step forward, but the abstract alone can't establish that the load-bearing estimates actually hold. I'd send it to referees, but I want the full text before I believe the sharp exponents.\n\nWhat's new: the abstract says only a sharp lower bound existed before, from the authors' own prior work, and this paper supplies the matching upper bound. That gives nonlinear SHE hitting probabilities the same two-sided control that Gaussian fields have had for years. The joint-density estimate for the vector (nonlinear solution, supremum of linear solution over a small rectangle) is also new, at least as presented. The authors are Dalang, Nualart, Pu—about as credible as it gets for Malliavin calculus applied to SPDEs. The framework is standard: write the density as an iterated Skorohod integral and estimate terms. No parameters being fitted, no circularity. The abstract is clear and honest about what the 'main effort' is.\n\nThe soft spot is exactly that main effort. The second component is a supremum over a rectangle—not a Malliavin-smooth functional. So either the density formula has been extended to handle this, or they regularize and pass to a limit. Either way, every term in the Skorohod formula must be finite and scale with the rectangle size to the sharp power. One divergent term or an extra log factor would sink sharpness. On an abstract-only read, that is a medium-risk gap, not a fatal one. There is no visible error in what's described, but there is also no evidence yet that the estimates work. The stress-test question about the Gaussian limit (zero nonlinearity) is the right test: the bound should reproduce the known Gaussian constant/exponent in that special case.\n\nBottom line: worth a serious referee. The result is important, the program is coherent, and the authors are the natural people to carry it out. But the referee needs to verify the Malliavin estimates carefully, especially the supremum regularization. I'd take it to a reading group once the full text is on arXiv, and I'd cite it if the proof checks out. For now: send it to peer review, but ask the referee to be brutal on the estimates.","headline":"A sharp upper bound for hitting probabilities of nonlinear SHE, from authors who know this area cold; the catch is the key Malliavin estimates are invisible from the abstract.","tokens_in":1783,"tokens_out":804,"would_cite":false,"duration_ms":11186,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60H07","60G60"],"pacs":[],"model":"deepseek-v4-flash","headline":"For nonlinear stochastic heat equations on the line, the paper proves a sharp upper bound on hitting probabilities that matches the known lower bound.","keywords":["stochastic heat equation","hitting probabilities","sharp bounds","Malliavin calculus","Skorohod integral","non-Gaussian random fields","SPDE","supremum density"],"falsifier":"Compute, in an explicitly solvable case such as additive space-time white noise, the probability that the nonlinear solution enters a ball of radius $\\varepsilon$ around a fixed point over a short time. The claimed sharp bound predicts a specific power of $\\varepsilon$; if the measured decay rate differs, the bound is not sharp. A more direct check is to evaluate the determinant of the Malliavin covariance of the pair (nonlinear solution, supremum of linear solution over a small rectangle): the proof requires it to remain bounded below by a constant times the rectangle's area, so a counterexam","tokens_in":751,"feed_emoji":"🎯","tokens_out":8978,"duration_ms":92524,"temperature":0.7,"pith_summary":"The paper establishes a sharp upper bound on the probability that the solution to a nonlinear stochastic heat equation on the line hits a fixed set. This bound matches, in order of magnitude, the sharp lower bound obtained in an earlier paper by two of the authors. Together, the two estimates put hitting probabilities for this non-Gaussian SPDE on the same footing as the classical sharp bounds for Gaussian random fields. The proof works by bounding the joint density of a two-dimensional random vector: the nonlinear solution at a point and the supremum of the linear stochastic heat equation over a small rectangle. The density is represented as an iterated Skorohod integral and each term is estimated with Malliavin calculus.","feed_headline":"Proved: sharp bound for hitting probabilities of nonlinear heat equation","feed_subtitle":"Two-sided sharp bounds now hold for hitting probabilities of nonlinear stochastic heat equations.","key_machinery":"The iterated Skorohod integral representation of the density of a locally nondegenerate random vector: a formula that expresses the joint density of such a vector as an iterated Skorohod integral. The proof applies this to the pair consisting of the nonlinear solution and the supremum of the linear solution over a small rectangle, then uses Malliavin calculus to estimate each term in the integral, so that the resulting density bound scales with the rectangle size as required.","core_discovery":"The central claim is a sharp upper bound for hitting probabilities of the solution to the nonlinear stochastic heat equation on the line. In the paper's technical form, the bound is obtained by controlling the joint probability density of the pair formed by the nonlinear solution and the supremum, over a small rectangle, of the corresponding linear solution. The density is expressed as an iterated Skorohod integral, valid when the pair is locally nondegenerate, and the main work is a Malliavin-calculus estimate of each term showing that the density is controlled in terms of the rectangle's size. This yields the upper bound that was missing, completing the two-sided sharp estimate.","pith_inferences":["Beyond the paper: the same two-sided bound should extend to systems of nonlinear stochastic heat equations and to higher spatial dimensions, because the order is set by the linear equation's small-scale scaling, which the proof isolates.","Beyond the paper: the mechanism suggests a general principle for semilinear SPDEs: the hitting-probability exponent is inherited from the linear part, so the Gaussian linear theory predicts the non-Gaussian exponent.","Beyond the paper: one could test the bound numerically in the additive-noise case, where the nonlinear solution is explicit, by comparing the estimated constant with the paper's density bound; a mismatch would indicate the constant is not sharp even if the power is."],"forward_implications":["The upper bound and the earlier lower bound now give two-sided estimates of the same order for hitting probabilities of nonlinear stochastic heat equations on the line.","These two-sided estimates match the sharp bounds long available for Gaussian random fields, so the nonlinearity does not change the order of the hitting probability.","The joint density bound for the pair (nonlinear solution, supremum of the linear solution over a small rectangle) is a reusable quantitative tool for non-Gaussian SPDEs, for instance in small-ball probability estimates.","The proof demonstrates that Malliavin-calculus density representations can handle random vectors that mix a point value and a supremum-type functional, going beyond the usual pointwise settings."],"supporting_citations":[],"fun_headline_variants":["Sharp upper bound completes two-sided hitting probability estimates","Nonlinear heat equation: hitting probability upper bound proved","Two-sided sharp bounds for nonlinear heat equation hitting probabilities","Hitting probabilities of nonlinear heat equation now sharply bounded"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The argument assumes the two-dimensional random vector is locally nondegenerate in the Malliavin sense, so the iterated Skorohod integral density formula is applicable; if that nondegeneracy fails, the density representation and the entire bound collapse.","fun_headline_variants_meta":{"raw":{"variants":["Sharp upper bound completes two-sided hitting probability estimates","Nonlinear heat equation: hitting probability upper bound proved","Two-sided sharp bounds for nonlinear heat equation hitting probabilities","Hitting probabilities of nonlinear heat equation now sharply bounded"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000839,"raw_usage":{"total_tokens":3474,"prompt_tokens":701,"completion_tokens":2773,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":445,"completion_tokens_details":{"reasoning_tokens":2711}},"tokens_in":445,"tokens_out":2773,"duration_ms":21595,"temperature":1.0,"reasoning_tokens":2711,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T19:42:20.831699+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, in an explicitly solvable case such as additive space-time white noise, the probability that the nonlinear solution enters a ball of radius $\\varepsilon$ around a fixed point over a short time. The claimed sharp bound predicts a specific power of $\\varepsilon$; if the measured decay rate differs, the bound is not sharp. A more direct check is to evaluate the determinant of the Malliavin covariance of the pair (nonlinear solution, supremum of linear solution over a small rectangle): the proof requires it to remain bounded below by a constant times the rectangle's area, so a counterexam","supporting_citations":[],"review_version":1}