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

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 Hölder 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--Itô integrals with deterministic kernels. As a consequence, the analysis avoids Malliavin calculus techniques that are typically required for non-Gaussian noises of Hermite rank $q \ge 2$.