Stochastic Burgers Equation Driven by a Hermite Sheet with Additive Noise: Existence, Uniqueness, and Regularity

Atef Lechiheb

arxiv: 2511.10463 · v3 · pith:PUMTYI7Jnew · submitted 2025-11-13 · 🧮 math.PR

Stochastic Burgers Equation Driven by a Hermite Sheet with Additive Noise: Existence, Uniqueness, and Regularity

Atef Lechiheb This is my paper

Pith reviewed 2026-05-25 07:59 UTC · model grok-4.3

classification 🧮 math.PR

keywords stochastic Burgers equationHermite sheetadditive noisemild solutionsHölder continuityself-similarityexistence and uniquenessWiener-Itô integrals

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The pith

The stochastic Burgers equation driven by an additive Hermite sheet admits unique mild solutions that are Hölder continuous under conditions on the Hurst parameters.

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

This paper proves existence and uniqueness of solutions to the stochastic Burgers equation with additive Hermite sheet noise of order q at least 1. It formulates the equation in mild form using the heat semigroup and applies a fixed-point argument in appropriate Banach spaces. Uniform moment estimates are derived when the Hurst parameters meet suitable restrictions, which then yield a continuous modification of the solution. The modification is Hölder continuous in both time and space, with the exponents fixed by the noise parameters, and the solution inherits anisotropic self-similarity from the driving sheet.

Core claim

Existence and uniqueness of mild solutions are established via a fixed-point argument in suitable Banach spaces. Under appropriate conditions on the Hurst parameters of the Hermite sheet, uniform moment estimates hold and imply that the solution admits a continuous modification that is Hölder continuous in time and space with exponents determined by those parameters. The solution also inherits an anisotropic self-similarity property with identified scaling exponents. The additive noise structure permits the stochastic convolution to be expressed through multiple Wiener-Itô integrals with deterministic kernels, bypassing Malliavin calculus.

What carries the argument

Fixed-point argument in Banach spaces applied to the mild formulation via the heat semigroup, where the stochastic convolution is defined by multiple Wiener-Itô integrals with deterministic kernels.

If this is right

Unique mild solutions exist for the equation under the stated conditions.
The solution possesses a Hölder continuous modification whose exponents are fixed by the Hurst parameters.
The solution inherits anisotropic self-similarity with explicitly identified scaling exponents.
Uniform moment bounds are available and support the regularity analysis.
The argument proceeds without Malliavin calculus for Hermite rank q at least 2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same fixed-point plus moment-estimate route may extend to other nonlinear SPDEs driven by additive Hermite sheets.
The explicit Hölder exponents could be used to select mesh sizes in numerical schemes that respect the regularity.
The inherited self-similarity may allow scaling arguments to be applied directly to long-time or large-space behavior of the solution.

Load-bearing premise

The Hurst parameters of the Hermite sheet must satisfy the restrictions that make the uniform moment estimates hold and render the fixed-point map a contraction.

What would settle it

A concrete counter-example in which the Hurst parameters lie outside the stated range and either the fixed-point iteration fails to converge or no continuous modification of the solution exists.

read the original abstract

We study the stochastic Burgers equation driven by an additive Hermite sheet of order $q \ge 1$. The equation is formulated in the mild sense using the heat semigroup, and existence and uniqueness of solutions are established via a fixed-point argument in suitable Banach spaces. Under appropriate conditions on the Hurst parameters of the Hermite sheet, we derive uniform moment estimates for the solution, which form the basis for the regularity analysis. We prove that the solution admits a continuous modification that is H\"older continuous in both time and space, with exponents determined by the Hurst parameters of the driving noise. In addition, we show that the solution inherits an anisotropic self-similarity property from the Hermite sheet, and we identify the corresponding scaling exponents. The additive noise structure allows the stochastic convolution to be defined through multiple Wiener--It\^o integrals with deterministic kernels. As a consequence, the analysis avoids Malliavin calculus techniques that are typically required for non-Gaussian noises of Hermite rank $q \ge 2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Standard mild-solution fixed-point result for Burgers with additive Hermite sheet noise, but the admissible Hurst ranges are left unspecified.

read the letter

The paper sets up the stochastic Burgers equation with additive Hermite sheet noise of order q at least 1, writes the mild solution via the heat semigroup, and runs a fixed-point argument in suitable Banach spaces to get existence and uniqueness. It then uses moment estimates to obtain Hölder continuity in time and space plus anisotropic self-similarity, with the exponents tied to the Hurst parameters of the sheet. The additive structure lets them define the stochastic convolution through multiple Wiener-Itô integrals with deterministic kernels, which removes the need for Malliavin calculus when q is at least 2. That is the main technical simplification and it is cleanly executed. The rest follows the usual pipeline in this corner of stochastic PDEs. The argument is not circular; it rests on the semigroup, the kernel integrals, and the contraction mapping theorem. What is missing is any explicit statement of the Hurst-parameter restrictions that make the moment bounds uniform and the fixed-point map contractive. The abstract only says “appropriate conditions,” so it is impossible to tell whether those conditions form a non-empty set for q greater than 1 or whether they force the Hölder exponents to zero. If the admissible region is empty or degenerate, both existence and the regularity claims fail. The stress-test note is right on this point. This is a paper for specialists already working on SPDEs driven by Hermite or Rosenblatt noises. A reader who wants to see how the mild-solution framework extends to this noise class will find the setup useful, but the lack of concrete parameter ranges limits how far one can trust the conclusions without the full estimates. I would send it to a referee who knows the literature on multiple stochastic integrals and fractional noises; the referee can check whether the conditions are realistic and whether the proofs actually close.

Referee Report

2 major / 1 minor

Summary. The paper studies the stochastic Burgers equation driven by additive noise given by a Hermite sheet of order q ≥ 1. The equation is formulated in mild form via the heat semigroup. Existence and uniqueness of solutions are proved by a fixed-point argument in suitable Banach spaces. Under appropriate conditions on the Hurst parameters, uniform moment estimates are obtained for the solution; these are used to establish the existence of a continuous modification that is Hölder continuous in time and space, with the Hölder exponents determined by the Hurst parameters of the driving noise. The solution is further shown to inherit an anisotropic self-similarity property from the Hermite sheet.

Significance. If the admissible set of Hurst parameters is non-empty, yields a contraction, and produces strictly positive Hölder exponents, the result would extend the theory of SPDEs with non-Gaussian additive noise of arbitrary Hermite rank q to the Burgers equation while avoiding Malliavin calculus. The representation of the stochastic convolution via multiple Wiener–Itô integrals with deterministic kernels is a methodological strength that simplifies the analysis.

major comments (2)

[Abstract] Abstract (paragraph on moment estimates and regularity analysis): the central claims of existence, uniqueness, and Hölder regularity all rest on 'appropriate conditions on the Hurst parameters' that are never stated explicitly. The uniform moment bounds on the stochastic convolution and the contraction property of the fixed-point map both depend on these conditions; without explicit inequalities involving the Hurst indices and q, it is impossible to verify that the admissible set is non-empty for q ≥ 2 or that the resulting Hölder exponents remain positive.
[Section on moment estimates] The section on moment estimates: the uniform moment estimates for the stochastic convolution (multiple Wiener–Itô integrals with deterministic kernels) are asserted only under the unspecified Hurst-parameter restrictions. Because these estimates are required both for the Banach-space contraction and for the subsequent Kolmogorov-type argument yielding Hölder continuity, the lack of verifiable parameter ranges renders the existence/uniqueness theorem and the regularity statement unverifiable as stated.

minor comments (1)

[Abstract] The abstract mentions that the analysis 'avoids Malliavin calculus techniques'; a brief sentence recalling why the additive structure permits the deterministic-kernel representation would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will revise the manuscript accordingly to improve clarity and verifiability.

read point-by-point responses

Referee: [Abstract] Abstract (paragraph on moment estimates and regularity analysis): the central claims of existence, uniqueness, and Hölder regularity all rest on 'appropriate conditions on the Hurst parameters' that are never stated explicitly. The uniform moment bounds on the stochastic convolution and the contraction property of the fixed-point map both depend on these conditions; without explicit inequalities involving the Hurst indices and q, it is impossible to verify that the admissible set is non-empty for q ≥ 2 or that the resulting Hölder exponents remain positive.

Authors: We agree that the abstract should state the conditions explicitly rather than referring only to 'appropriate conditions.' In the revised version we will insert the precise inequalities on the Hurst parameters (involving H_t, the spatial Hurst indices, and q) directly into the abstract. These inequalities are already derived and used in the body of the paper to obtain the moment bounds and the contraction; making them visible in the abstract will allow immediate verification that the admissible set is non-empty for every q ≥ 1 and that the resulting Hölder exponents are strictly positive. revision: yes
Referee: [Section on moment estimates] The section on moment estimates: the uniform moment estimates for the stochastic convolution (multiple Wiener–Itô integrals with deterministic kernels) are asserted only under the unspecified Hurst-parameter restrictions. Because these estimates are required both for the Banach-space contraction and for the subsequent Kolmogorov-type argument yielding Hölder continuity, the lack of verifiable parameter ranges renders the existence/uniqueness theorem and the regularity statement unverifiable as stated.

Authors: We accept that the moment-estimates section must display the Hurst-parameter restrictions explicitly. We will add a clearly labeled assumption block (or a dedicated paragraph) at the beginning of the section that states the precise inequalities required for the L^p bounds on the multiple Wiener–Itô integrals. With these inequalities stated, the contraction-mapping argument and the subsequent Kolmogorov criterion become directly verifiable from the text. revision: yes

Circularity Check

0 steps flagged

No circularity; standard fixed-point existence proof conditional on Hurst-parameter restrictions.

full rationale

The paper establishes mild solutions to the stochastic Burgers equation via the heat semigroup and a Banach fixed-point argument on the stochastic convolution (defined by multiple Wiener-Itô integrals with deterministic kernels). Uniform moment estimates and subsequent Hölder regularity are asserted only under unspecified 'appropriate conditions' on the Hurst parameters; these conditions are external hypotheses required for the contraction mapping and moment bounds to hold, not quantities derived or fitted inside the paper. No self-definitional loops, no fitted inputs renamed as predictions, and no load-bearing self-citations appear in the derivation chain. The argument is therefore self-contained against external benchmarks (semigroup theory, fixed-point theorem, Wiener-Itô calculus) once the parameter restrictions are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard functional-analytic tools and domain assumptions about the noise; no free parameters are fitted to data and no new entities are postulated.

axioms (2)

standard math The Banach fixed-point theorem applies once the map is shown to be a contraction on the chosen space of processes
Invoked for existence and uniqueness in the mild formulation.
domain assumption The stochastic convolution admits a representation as multiple Wiener-Itô integrals with deterministic kernels when the noise is additive
Used to bypass Malliavin calculus for q ≥ 2.

pith-pipeline@v0.9.0 · 5709 in / 1494 out tokens · 40526 ms · 2026-05-25T07:59:42.554014+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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[1] [1]

and Zabczyk, J

Da Prato, G. and Zabczyk, J. (1992).Stochastic Equations in Infinite Dimensions. Cam- bridge University Press

work page 1992

[2] [2]

Hairer, M. (2013). Solving the KPZ equation.Annals of Mathematics, 178:559–664

work page 2013

[3] [3]

Hairer, M. (2014). A theory of regularity structures.Inventiones Mathematicae, 198:269– 504

work page 2014

[4] [4]

and Giacomin, G

Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations from particle systems.Communications in Mathematical Physics, 183:571–607

work page 1997

[5] [5]

Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosen- blatt process.Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 31:287– 302

work page 1975

[6] [6]

Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 50:53–83

work page 1979

[7] [7]

Tudor, C. A. (2013).Analysis of Variations for Self-similar Processes. Springer

work page 2013

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Balan, R. M. and Tudor, C. A. (2014). The stochastic heat equation with fractional-colored noise.Journal of Theoretical Probability, 27:1–34

work page 2014

[9] [9]

and Tudor, C

Slaoui, M. and Tudor, C. A. (2019). Stochastic heat equation driven by fractional colored noise.Infinite Dimensional Analysis, Quantum Probability and Related Topics, 22(04):1950022. DOI: 10.1142/S021902571950022X

work page doi:10.1142/s021902571950022x 2019

[10] [10]

Čoupek, P., Maslowski, B., and Šiška, D. (2020). Stochastic evolution equations with Rosenblatt noise.Stochastic Processes and their Applications, 130(2):694–724. DOI: 10.1016/j.spa.2020.01.004

work page doi:10.1016/j.spa.2020.01.004 2020

[11] [11]

Čoupek, P

P. Čoupek, P. Kříž and M. Svoboda. On the1/H-variation of the divergence integral with respect to a Hermite process.Preprint, 2025.https://arxiv.org/abs/2509.11737v1

work page arXiv 2025

[12] [12]

(1997).Gaussian Hilbert Spaces

Janson, S. (1997).Gaussian Hilbert Spaces. Cambridge University Press

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(2006).The Malliavin Calculus and Related Topics

Nualart, D. (2006).The Malliavin Calculus and Related Topics. Springer. 22

work page 2006

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and Taqqu, M

Peccati, G. and Taqqu, M. S. (2011).Wiener Chaos: Moments, Cumulants and Diagrams. Springer

work page 2011

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Hu, Y., Nualart, D., and Song, J. (2011). Feynman–Kac formula for heat equation driven by fractional white noise.Annals of Probability, 39(1):291–326. DOI: 10.1214/10-AOP547

work page doi:10.1214/10-aop547 2011

[16] [16]

Gubinelli, M., Imkeller, P., and Perkowski, N. (2015). Paracontrolled distributions and singular PDEs.Forum of Mathematics, Pi, 3:e6. 23

work page 2015