Invariance of $ϕ^4$ measure under nonlinear wave and Schrödinger equations on the plane
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914249048064000 |
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| author | Barashkov, Nikolay Laarne, Petri |
| author_facet | Barashkov, Nikolay Laarne, Petri |
| contents | We show almost sure wellposedness of mild solution to the cubic nonlinear wave equation in a weighted Besov space over $\mathbb R^2$. To achieve this, we show that any weak limit of $ϕ^4$ measures on increasing tori is invariant under the equation. We review and slightly simplify the periodic theory and the construction of the weak limit measure, and then use finite speed of propagation to reduce the infinite-volume case to the previous setup. Our argument also gives a weaker invariance result on the nonlinear Schrödinger equation in the same setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_16111 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Invariance of $ϕ^4$ measure under nonlinear wave and Schrödinger equations on the plane Barashkov, Nikolay Laarne, Petri Analysis of PDEs Mathematical Physics Probability 35L71, 60H30 (Primary) 35Q55, 60H15, 81T08 (Secondary) We show almost sure wellposedness of mild solution to the cubic nonlinear wave equation in a weighted Besov space over $\mathbb R^2$. To achieve this, we show that any weak limit of $ϕ^4$ measures on increasing tori is invariant under the equation. We review and slightly simplify the periodic theory and the construction of the weak limit measure, and then use finite speed of propagation to reduce the infinite-volume case to the previous setup. Our argument also gives a weaker invariance result on the nonlinear Schrödinger equation in the same setting. |
| title | Invariance of $ϕ^4$ measure under nonlinear wave and Schrödinger equations on the plane |
| topic | Analysis of PDEs Mathematical Physics Probability 35L71, 60H30 (Primary) 35Q55, 60H15, 81T08 (Secondary) |
| url | https://arxiv.org/abs/2211.16111 |