Quasi invariant Gaussian measures for the nonlinear Schrödinger equation on $\mathbb T^2$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908712075001856 |
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| author | Tolomeo, Leonardo Visciglia, Nicola |
| author_facet | Tolomeo, Leonardo Visciglia, Nicola |
| contents | We study the transport of Gaussian measures under the flow of the 2-dimensional defocusing Schrödinger equation $i \partial_t u + Δu = |u|^{2k} u$ posed on $\mathbb T^2$. In particular, we show that the Gaussian measures with inverse covariance $\|u\|_{H^s}^2$, are quasi-invariant under the flow for $s>2$. Moreover, we show that the Radon-Nykodim density belongs to every $L^p$ space, locally in space. The proof relies on the physical-space energies introduced in [52], as well as a new abstract quasi-invariance argument that allows us to combine space-time estimates, along the flow with probabilistic bounds on the support of the measure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_13113 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi invariant Gaussian measures for the nonlinear Schrödinger equation on $\mathbb T^2$ Tolomeo, Leonardo Visciglia, Nicola Analysis of PDEs 35Q55, 35R60, 37A40, 60H30 We study the transport of Gaussian measures under the flow of the 2-dimensional defocusing Schrödinger equation $i \partial_t u + Δu = |u|^{2k} u$ posed on $\mathbb T^2$. In particular, we show that the Gaussian measures with inverse covariance $\|u\|_{H^s}^2$, are quasi-invariant under the flow for $s>2$. Moreover, we show that the Radon-Nykodim density belongs to every $L^p$ space, locally in space. The proof relies on the physical-space energies introduced in [52], as well as a new abstract quasi-invariance argument that allows us to combine space-time estimates, along the flow with probabilistic bounds on the support of the measure. |
| title | Quasi invariant Gaussian measures for the nonlinear Schrödinger equation on $\mathbb T^2$ |
| topic | Analysis of PDEs 35Q55, 35R60, 37A40, 60H30 |
| url | https://arxiv.org/abs/2512.13113 |