Stochastic Burgers Equation Driven by Hermite Sheet: Existence, Uniqueness, and Regularity Properties

Abstract: We analyze the stochastic Burgers' equation driven by a Hermite sheet of order ( q \geq 1 ), establishing well-posedness of mild solutions via fixed-point arguments 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, with explicit scaling exponents connecting temporal and spatial dimensions. Moment estimates are derived uniformly in space and time, providing the foundation for the regularity analysis. The work develops stochastic integration theory with respect to Hermite sheets 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.