Stochastic Burgers equation driven by multiplicative Rosenblatt noise: local existence, uniqueness and regularity

TL;DR

This paper establishes local existence, uniqueness, and regularity results for the stochastic Burgers equation driven by multiplicative Rosenblatt noise, a non-Gaussian long-memory process, using advanced Malliavin calculus techniques.

Contribution

It provides the first rigorous analysis of a nonlinear SPDE driven by Rosenblatt noise, including sharp regularity and moment bounds.

Findings

Proved local existence and uniqueness of solutions.

Established sharp spatial and temporal H"older regularity.

Derived uniform moment bounds for solutions.

Abstract

We study the stochastic Burgers equation driven by a multiplicative Rosenblatt noise with Hurst parameter $H \in (1/2,1)$. Using a fixed-point argument in a Malliavin--Sobolev space that controls the solution and its first two Malliavin derivatives, we prove local existence and uniqueness of a mild solution. We establish uniform moment bounds of all orders and prove H\"older regularity: spatial H\"older exponent $\gamma < 1/2$ and temporal H\"older exponent $\alpha < H-1/2$, which are shown to be sharp by a lower bound for the linearized equation. The proof relies on sharp estimates of the heat kernel in the reproducing kernel Hilbert space $\cH$ of the Rosenblatt process, on Meyer's inequalities for moment bounds, and on a careful analysis of the Skorohod integral with respect to the Rosenblatt process. These results provide a rigorous foundation for the study of nonlinear SPDEs driven by non-Gaussian long-memory noise.