Central limit theorem for the focusing $Φ^4$-measure in the infinite volume limit
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910968236212224 |
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| author | Seong, Kihoon Sosoe, Philippe |
| author_facet | Seong, Kihoon Sosoe, Philippe |
| contents | We study the fluctuations of the focusing $Φ^4$-measure on the one-dimensional torus in the infinite volume limit. This measure is an invariant Gibbs measure for the nonlinear Schrödinger equation. It had previously been shown by B. Rider that the measure is strongly concentrated around a family of minimizers of the Hamiltonian associated with the measure. These exhibit increasingly sharp spatial concentration, resulting in a trivial limit to first order. We study the fluctuations around this soliton manifold. We show that the scaled field under the Gibbs measure converges to white noise in the limit, identifying the next order fluctuations predicted by B. Rider. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_07840 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Central limit theorem for the focusing $Φ^4$-measure in the infinite volume limit Seong, Kihoon Sosoe, Philippe Probability Mathematical Physics Analysis of PDEs We study the fluctuations of the focusing $Φ^4$-measure on the one-dimensional torus in the infinite volume limit. This measure is an invariant Gibbs measure for the nonlinear Schrödinger equation. It had previously been shown by B. Rider that the measure is strongly concentrated around a family of minimizers of the Hamiltonian associated with the measure. These exhibit increasingly sharp spatial concentration, resulting in a trivial limit to first order. We study the fluctuations around this soliton manifold. We show that the scaled field under the Gibbs measure converges to white noise in the limit, identifying the next order fluctuations predicted by B. Rider. |
| title | Central limit theorem for the focusing $Φ^4$-measure in the infinite volume limit |
| topic | Probability Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2411.07840 |