Renormalization of a 1d quadratic Schr{ö}dinger model with additive noise

The study is devoted to the interpretation and wellposedness of the stochastic NLS model \begin{equation*} (\imath \partial_t-Δ)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 $0<2H_0+H_1\le 2$, the equation cannot be interpreted in a (classical) functional sense.\ \indent Our investigations then focus on the rough regime corresponding to the condition $\frac{7}{4}<2H_0+H_1\le 2$. In this specific case, we exhibit an $\mathit{\text{explicit}}$ 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.