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

TL;DR

This paper investigates the stochastic Burgers equation driven by a Hermite sheet with additive noise, establishing existence, uniqueness, and regularity of solutions under specific conditions on the Hurst parameters, and highlighting the advantages of additive noise for analysis.

Contribution

It provides a novel framework for analyzing nonlinear SPDEs driven by Hermite sheets with additive noise, avoiding complex Malliavin calculus techniques used for multiplicative noise.

Findings

Proved existence and uniqueness of solutions under Hurst parameter conditions.

Established spatial and temporal Hölder regularity of solutions.

Derived explicit self-similarity and scaling properties of the solutions.

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 H\"older 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.