The study is devoted to the interpretation and wellposedness of the stochastic NLS model \begin{ equation*} (\imath \partial_t-\Delta)u=|u|^2+\dot{ B}, \quad u_0=0,\quad \quad t\in \mathbb{ R}, \ x\in \mathbb{ T}, \end{ equation*} where $\dot{ B}$ stands for a space-time fractional noise with index $H=(H_0,H_1)$ in a subset of $(0,1)^{ 2}$. We first establish that in the situation where $0<2H_0+H_1\leq 2$, the equation cannot be interpreted in a (classical) functional sense.\\ \indent Our investigations then focus on the rough regime corresponding to the condition $\frac74<2H_0+H_1\leq 2$. In this specific case, we exhibit an \textit{ explicit} renormalization procedure allowing to restore the (local) convergence of the approximated solutions. We follow a pathwise-type approach emphasizing the distinction between the stochastic objects at the core of the dynamics and the general deterministic machinery.
Categories: math.AP math.PR