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

Atef Lechiheb

TL;DR

This work analyzes the stochastic Burgers equation driven by a Hermite sheet with additive noise, establishing local well-posedness of mild solutions via a fixed-point method under a Hurst-parameter condition that becomes more stringent with higher Hermite rank. It develops a deterministic-integrand stochastic integration framework for Hermite sheets, yielding uniform moment bounds and Hölder regularity dictated by the temporal and spatial Hurst parameters. The authors also show the solution inherits the Hermite sheet's self-similarity under anisotropic scaling, linking the SPDE dynamics to the underlying non-Gaussian long-range dependence. By focusing on the additive case, the paper avoids Malliavin-calculus complexities while providing a solid foundation for future extensions to multiplicative noise and renormalization techniques in non-Gaussian settings.

Abstract

We study the stochastic Burgers equation driven by a Hermite sheet of order \( q \geq 1 \) with \textbf{additive noise}, establishing the well-posedness of mild solutions via a fixed-point argument in suitable Banach spaces. Under appropriate conditions on the Hurst parameters \( \mathbf{H} = (H_0, H_1, \dots, H_d) \in (1/2, 1)^{d+1} \), we prove existence and uniqueness of solutions through a Picard iteration scheme. The solution exhibits spatial and temporal Hölder regularity, with exponents determined by the Hurst parameters of the driving noise. Furthermore, we demonstrate that the solution inherits the self-similarity property from the Hermite sheet, providing explicit scaling exponents. Uniform moment estimates in space and time are derived, forming the foundation for the regularity analysis. The additive noise formulation allows us to use the standard Wiener integral construction for Hermite processes, thereby avoiding the technical complications of Malliavin calculus required for multiplicative noise. This restriction is mathematically justified as it circumvents the need for Malliavin derivative bounds essential for random integrands with Hermite processes of order \(q \geq 2\), a key difficulty highlighted in recent literature. The work develops the stochastic integration theory with respect to Hermite sheets for deterministic integrands and establishes a complete framework for analyzing nonlinear SPDEs with non-Gaussian noise, contributing to the understanding of stochastic systems with long-range dependence and non-Gaussian fluctuations.

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

TL;DR

This work analyzes the stochastic Burgers equation driven by a Hermite sheet with additive noise, establishing local well-posedness of mild solutions via a fixed-point method under a Hurst-parameter condition that becomes more stringent with higher Hermite rank. It develops a deterministic-integrand stochastic integration framework for Hermite sheets, yielding uniform moment bounds and Hölder regularity dictated by the temporal and spatial Hurst parameters. The authors also show the solution inherits the Hermite sheet's self-similarity under anisotropic scaling, linking the SPDE dynamics to the underlying non-Gaussian long-range dependence. By focusing on the additive case, the paper avoids Malliavin-calculus complexities while providing a solid foundation for future extensions to multiplicative noise and renormalization techniques in non-Gaussian settings.

Abstract

We study the stochastic Burgers equation driven by a Hermite sheet of order with \textbf{additive noise}, establishing the well-posedness of mild solutions via a fixed-point argument in suitable Banach spaces. Under appropriate conditions on the Hurst parameters \( \mathbf{H} = (H_0, H_1, \dots, H_d) \in (1/2, 1)^{d+1} \), we prove existence and uniqueness of solutions through a Picard iteration scheme. The solution exhibits spatial and temporal Hölder regularity, with exponents determined by the Hurst parameters of the driving noise. Furthermore, we demonstrate that the solution inherits the self-similarity property from the Hermite sheet, providing explicit scaling exponents. Uniform moment estimates in space and time are derived, forming the foundation for the regularity analysis. The additive noise formulation allows us to use the standard Wiener integral construction for Hermite processes, thereby avoiding the technical complications of Malliavin calculus required for multiplicative noise. This restriction is mathematically justified as it circumvents the need for Malliavin derivative bounds essential for random integrands with Hermite processes of order , a key difficulty highlighted in recent literature. The work develops the stochastic integration theory with respect to Hermite sheets for deterministic integrands and establishes a complete framework for analyzing nonlinear SPDEs with non-Gaussian noise, contributing to the understanding of stochastic systems with long-range dependence and non-Gaussian fluctuations.

Paper Structure

This paper contains 41 sections, 6 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Novelty and Challenges
  3. Technical Justification for Additive Noise
  4. Advantages of the Additive Noise Framework
  5. Main Results
  6. Organization of the Paper
  7. Preliminaries
  8. Multiple Wiener--Itô Integrals and Wiener Chaos
  9. Hermite Processes and Hermite Sheets
  10. Stochastic Integration with Respect to the Hermite Sheet
  11. Formulation of the Stochastic Burgers' Equation with Additive Noise
  12. Mild Formulation for Additive Noise
  13. Functional framework.
  14. Assumptions for Additive Noise
  15. Contrast with multiplicative noise.
  16. ...and 26 more sections

Key Result

Proposition 4.1

Under Assumption assump:additive-data(b,c), for every fixed $t > 0$ and $\mathbf{x} \in \mathbb{R}^d$, the deterministic function belongs to the Hilbert space $\mathcal{H}^{\otimes q}$ associated with the Hermite noise $Z^{q,\mathbf{H}}$. Consequently, the Wiener integral is well-defined as a multiple Wiener-Itô integral $I_q(\tilde{f})$, where $\tilde{f}$ is the symmetrization of $f^{\otimes q}

Theorems & Definitions (25)

  • Definition 2.1: Hermite process
  • Definition 2.2: Hermite sheet
  • Definition 2.3: Covariance Hilbert space
  • Definition 2.4: Wiener integral for deterministic integrands
  • Remark 2.5: Identification of $\mathcal{H}^{\otimes q}$
  • Definition 3.1: Mild Solution for Additive Noise
  • Remark 3.3: Interpretation and optimality of the Hurst condition
  • Proposition 4.1: Well-definedness of Stochastic Convolution
  • proof : Proof sketch
  • Remark 4.2: Contrast with multiplicative noise
  • ...and 15 more