Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line

David Nualart; Fei Pu; Robert C. Dalang

REVIEW 3 major objections 3 minor 1 cited by

Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For nonlinear stochastic heat equations on the line, the paper proves a sharp upper bound on hitting probabilities that matches the known lower bound.

desk verdict 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. read the letter →

arxiv 2508.11859 v1 pith:PUCBVBYT submitted 2025-08-16 math.PR

Robert C. Dalang , David Nualart , Fei Pu This is my paper · ORCID

classification math.PR MSC 60H1560H0760G60

keywords stochasticheatequationhittingprobabilitiessharpboundsMalliavincalculusSkorohodintegralnon-GaussianrandomfieldsSPDEsupremumdensity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What'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.

The 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.

Bottom 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.

Referee Report

3 major / 3 minor

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.

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 (3)

[Abstract] 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.
[Abstract] 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.
[Abstract] 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.

minor comments (3)

[Abstract] 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.
[Abstract] 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.
[Abstract] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the sharp upper bound is derived from standard Malliavin calculus and compared to an independent prior lower bound.

full rationale

The abstract describes a proof of a sharp upper bound on hitting probabilities for the nonlinear stochastic heat equation. The load-bearing estimate is a bound on the joint density of (u(t,x), sup_R v), obtained via an iterated Skorohod integral density formula for locally nondegenerate random vectors. This formula is a standard result of Malliavin calculus, not a self-cited or fitted input. The only self-citation is to the authors' previous sharp lower bound, which serves as the external benchmark for sharpness, not as an ingredient in the upper-bound derivation. There are no fitted parameters and no result defined into existence. Although the full text is unavailable, nothing in the abstract suggests that the argument reduces to its own assumptions. The nondegeneracy condition and the finiteness/scaling of the Malliavin estimates are genuine technical premises; they are not equivalent to the claimed probability inequality. Therefore no circularity is identifiable.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

This is a pure mathematics paper: there are no fitted parameters, no data, and no invented entities. The ledger therefore records the mathematical premises the central claim rests on: standard well-posedness of the nonlinear SHE, the local nondegeneracy of the joint law (nonlinear solution, supremum of linear solution), the Malliavin calculus density representation used as the starting point, and the authors' prior sharp lower bound used to define what 'sharp' means. The local nondegeneracy item is the load-bearing one: the abstract identifies the Malliavin estimates of the density formula terms as the main effort.

assumptions (4)

domain assumption Unique solution to the nonlinear stochastic heat equation system with the regularity required for hitting probability analysis.
The abstract studies hitting probabilities of the solution to a nonlinear system of SHEs; existence, uniqueness, and Hölder regularity are standard SPDE inputs, not proved in the abstract.
domain assumption The 2D random vector (nonlinear SHE solution at a point, supremum over a small rectangle of the linear SHE solution) is locally nondegenerate.
The abstract states the proof 'makes use of a formula that expresses the density of a locally nondegenerate random vector'; the local nondegeneracy of this specific joint law is the enabling condition for the density formula.
standard math The iterated Skorohod integral density formula and the Malliavin integration-by-parts identities hold as stated.
Taken from the Malliavin calculus literature (Nualart's framework); the paper applies this known representation rather than proving it.
domain assumption The previously obtained sharp lower bound on hitting probabilities (previous paper by two of the authors) is correct.
The abstract frames the new result as 'the corresponding sharp upper bound', so sharpness is benchmarked against that lower bound; the upper bound proof itself is independent of it.

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Pith. "Pith review of Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line." pith.science (2026). https://pith.science/paper/PUCBVBYT

@misc{pith2026250811859,
  author       = {Pith},
  title        = {Pith review of: Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PUCBVBYT}},
  note         = {Machine review of arXiv:2508.11859}
}

abstract

For Gaussian random fields with values in $\mathbb{R}^d$, sharp upper and lower bounds on the probability of hitting a fixed set have been available for many years. These apply in particular to the solutions of systems of linear SPDEs. For non-Gaussian random fields, the available bounds are less sharp. For nonlinear systems of stochastic heat equations, a sharp lower bound was obtained in a previous paper by two of the authors. Here, we obtain the corresponding sharp upper bound. The proof requires a bound on the joint probability density function of a two-dimensional random vector whose components are the solution to the {\em nonlinear} stochastic heat equation and the supremum over a small rectangle of the solution to the {\em linear} stochastic heat equation, in terms of the size of the rectangle. This bound makes use of a formula that expresses the density of a {\em locally nondegenerate} random vector as an iterated Skorohod integral. The main effort is to estimate, using Malliavin calculus, each of the terms that arise from this formula.

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