On the 1d stochastic Schr{ö}dinger product

Abstract

We exhibit various restrictions about the wellposedness of the Schr{\''o}dinger product $$\cl:z \longmapsto -\imath \int\_0^t e^{\imath s {\cop \partial^2\_x}}\big( z\_s\cdot Ψ\_s\big) ds $$ where $\hat{I}¨$ refers to the so-called linear solution of the stochastic Schr{\''o}dinger problem. We focus more specifically on the case where $\hat{I}¨$ 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{$u=\hat{I}¨+z$}).