Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.20281 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908682435952640 |
|---|---|
| author | Deya, Aurélien |
| author_facet | Deya, Aurélien |
| contents | 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 $Ψ$ refers to the so-called linear solution of the stochastic Schr{\''o}dinger problem. We focus more specifically on the case where $Ψ$ 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 $H>\frac12$. \smallskip As an consequence of our analysis, we obtain that if $H$ is close to $\frac12$ (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{$u=Ψ+z$}). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20281 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the 1d stochastic Schr{ö}dinger product Deya, Aurélien Analysis of PDEs Probability 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 $Ψ$ refers to the so-called linear solution of the stochastic Schr{\''o}dinger problem. We focus more specifically on the case where $Ψ$ 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 $H>\frac12$. \smallskip As an consequence of our analysis, we obtain that if $H$ is close to $\frac12$ (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{$u=Ψ+z$}). |
| title | On the 1d stochastic Schr{ö}dinger product |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2310.20281 |