On the 1d stochastic Schrödinger product
Abstract
We exhibit various restrictions about the wellposedness of the Schrödinger product $$\mathcal{L}: z \longmapsto -\imath \int_0^t e^{\imath \Delta s}\big( z_s\cdot \Psi_s\big) ds $$ where $\Psi$ refers to the so-called linear solution of the stochastic Schrdinger problem. We focus more specifically on the case where $\Psi$ satisfies $$(\imath \partial_t-\Delta)\Psi=\dot{B}, \quad \Psi_0=0,\quad \quad t\in \mathbb{R}, \ x\in \mathbb{T},$$where $\dot{B}$ is a white noise in space with fractional time covariance of index $H>\frac12$. As an important 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 $$(\imath \partial_t-\Delta)u=\lambda\, u^{p} \overline{u}^q+\dot{B}, \quad u_0=0,\quad \quad \lambda \in \{-1,1\}, \ p,q\geq 1, \ t\in \mathbb{R}, \ x\in \mathbb{T},$$using only a first-order expansion of the solution ("$u=\Psi+z$").
- Publication:
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arXiv e-prints
- Pub Date:
- October 2023
- DOI:
- 10.48550/arXiv.2310.20281
- arXiv:
- arXiv:2310.20281
- Bibcode:
- 2023arXiv231020281D
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematics - Probability