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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2023
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| Accès en ligne: | https://arxiv.org/abs/2304.03114 |
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| _version_ | 1866914176367067136 |
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| author | Deya, Aurélien Fukuizumi, Reika Thomann, Laurent |
| author_facet | Deya, Aurélien Fukuizumi, Reika Thomann, Laurent |
| contents | 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 $0<2H_0+H_1\leq 2$, the equation cannot be interpreted in a (classical) functional sense.\\ \indent Our investigations then focus on the rough regime corresponding to the condition $\frac74<2H_0+H_1\leq 2$. In this specific case, we exhibit an \textit{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_03114 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Renormalization of a 1d quadratic Schr{ö}dinger model with additive noise Deya, Aurélien Fukuizumi, Reika Thomann, Laurent Analysis of PDEs Probability 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 $0<2H_0+H_1\leq 2$, the equation cannot be interpreted in a (classical) functional sense.\\ \indent Our investigations then focus on the rough regime corresponding to the condition $\frac74<2H_0+H_1\leq 2$. In this specific case, we exhibit an \textit{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. |
| title | Renormalization of a 1d quadratic Schr{ö}dinger model with additive noise |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2304.03114 |