Existence, uniqueness, and universality of global dynamics for the fractional hyperbolic $Φ^4_3$-model
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arXiv
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| Format: | Preprint |
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2023
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| author | Liu, Ruoyuan Tzvetkov, Nikolay Wang, Yuzhao |
| author_facet | Liu, Ruoyuan Tzvetkov, Nikolay Wang, Yuzhao |
| contents | We study the fractional $Φ^4_3$-measure (with order $α> 1$) and the dynamical problem of its canonical stochastic quantization: the three-dimensional stochastic damped fractional nonlinear wave equation with a cubic nonlinearity, also called the fractional hyperbolic $Φ^4_3$-model. We first construct the fractional $Φ^4_3$-measure via the variational approach by Barashkov-Gubinelli (2020). When $α\leq \frac{9}{8}$, this fractional $Φ^4_3$-measure turns out to be singular with respect to the base Gaussian measure. We then prove almost sure global well-posedness of the fractional hyperbolic $Φ^4_3$-model and invariance of the fractional $Φ^4_3$-measure for all $α> 1$ by further developing the globalization framework due to Oh-Okamoto-Tolomeo (2024) on the hyperbolic $Φ^3_3$-model. Furthermore, when $α> \frac{9}{8}$ , we prove weak universality of the fractional hyperbolic $Φ^4_3$-model by utilizing the convergence of Gibbs measures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_00543 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Existence, uniqueness, and universality of global dynamics for the fractional hyperbolic $Φ^4_3$-model Liu, Ruoyuan Tzvetkov, Nikolay Wang, Yuzhao Analysis of PDEs Probability 60H15, 81T08, 35L71, 35R60 We study the fractional $Φ^4_3$-measure (with order $α> 1$) and the dynamical problem of its canonical stochastic quantization: the three-dimensional stochastic damped fractional nonlinear wave equation with a cubic nonlinearity, also called the fractional hyperbolic $Φ^4_3$-model. We first construct the fractional $Φ^4_3$-measure via the variational approach by Barashkov-Gubinelli (2020). When $α\leq \frac{9}{8}$, this fractional $Φ^4_3$-measure turns out to be singular with respect to the base Gaussian measure. We then prove almost sure global well-posedness of the fractional hyperbolic $Φ^4_3$-model and invariance of the fractional $Φ^4_3$-measure for all $α> 1$ by further developing the globalization framework due to Oh-Okamoto-Tolomeo (2024) on the hyperbolic $Φ^3_3$-model. Furthermore, when $α> \frac{9}{8}$ , we prove weak universality of the fractional hyperbolic $Φ^4_3$-model by utilizing the convergence of Gibbs measures. |
| title | Existence, uniqueness, and universality of global dynamics for the fractional hyperbolic $Φ^4_3$-model |
| topic | Analysis of PDEs Probability 60H15, 81T08, 35L71, 35R60 |
| url | https://arxiv.org/abs/2311.00543 |