Sharp quasi-invariance threshold for the cubic Szegő equation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929325280854016 |
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| author | Coe, James Tolomeo, Leonardo |
| author_facet | Coe, James Tolomeo, Leonardo |
| contents | We consider the 1-dimensional cubic Szegő equation with data distributed according to the Gaussian measure with inverse covariance operator $(1-\partial_x^2)^\frac s2$, where $s>\frac12$. We show that, for $s>1$, this measure is quasi-invariant under the flow of the equation, while for $s<1$, $s\neq \frac34$, the transported measure and the initial Gaussian measure are mutually singular for almost every time. This is the first observation of a transition from quasi-invariance to singularity in the context of the transport of Gaussian measures under the flow of Hamiltonian PDEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14950 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp quasi-invariance threshold for the cubic Szegő equation Coe, James Tolomeo, Leonardo Analysis of PDEs Probability 35R60, 37A40, 60H30 We consider the 1-dimensional cubic Szegő equation with data distributed according to the Gaussian measure with inverse covariance operator $(1-\partial_x^2)^\frac s2$, where $s>\frac12$. We show that, for $s>1$, this measure is quasi-invariant under the flow of the equation, while for $s<1$, $s\neq \frac34$, the transported measure and the initial Gaussian measure are mutually singular for almost every time. This is the first observation of a transition from quasi-invariance to singularity in the context of the transport of Gaussian measures under the flow of Hamiltonian PDEs. |
| title | Sharp quasi-invariance threshold for the cubic Szegő equation |
| topic | Analysis of PDEs Probability 35R60, 37A40, 60H30 |
| url | https://arxiv.org/abs/2404.14950 |