On the 1d stochastic Schr{ö}dinger product

We exhibit various restrictions about the wellposedness of the Schr{''o}dinger product $$\text{cl}:z⟼-\mathrm{ı}\int \mathrm{\_}{0}^t{e}^{\mathrm{ı}s\text{cop}{\mathrm{\partial }}^2\mathrm{\_}x}(z\mathrm{\_}s\cdot \mathrm{\Psi }\mathrm{\_}s)ds$$ where $\mathrm{\Psi }$ refers to the so-called linear solution of the stochastic Schr{''o}dinger problem. We focus more specifically on the case where $\mathrm{\Psi }$ satisfies \begin{equation}\label{starting-equation-abstract} (\imath \partial_t-\partial^2_x)Ψ=\dot{B}, \quad Ψ_0=0,\quad \quad t\in \R, \ x\in $\mathbb{T}$, \end{equation} where $\dot{B}$ is a white noise in space with fractional time covariance of index $H>\frac{1}{2}$. \smallskip As an consequence of our analysis, we obtain that if $H$ is close to $\frac{1}{2}$ (that is $\dot{B}$ is close to a space-time white noise), then it is essentially impossible to treat the stochastic NLS problem \begin{equation*} (\imath \partial_t-\partial^2_x)u= |u|^2+\dot{B}, \quad u_0=0,\quad \quad t\in \R, \ x\in $\mathbb{T}$, \end{equation*} using only a first-order expansion of the solution (\enquote{$u=\mathrm{\Psi }+z$}).