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.
Categories: math.PR