Hyperbolic $P(Φ)_2$-model on the plane
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arXiv
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2022
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| author | Oh, Tadahiro Tolomeo, Leonardo Wang, Yuzhao Zheng, Guangqu |
| author_facet | Oh, Tadahiro Tolomeo, Leonardo Wang, Yuzhao Zheng, Guangqu |
| contents | We study the hyperbolic $Φ^{k+1}_2$-model on the plane. By establishing coming down from infinity for the associated stochastic nonlinear heat equation (SNLH) on the plane, we first construct a $Φ^{k+1}_2$-measure on the plane as a limit of the $Φ^{k+1}_2$-measures on large tori. We then study the canonical stochastic quantization of the $Φ^{k+1}_2$-measure on the plane thus constructed, namely, we study the defocusing stochastic damped nonlinear wave equation forced by an additive space-time white noise (= the hyperbolic $Φ^{k+1}_2$-model) on the plane. In particular, by taking a limit of the invariant Gibbs dynamics on large tori constructed by the first two authors with Gubinelli and Koch (2021), we construct invariant Gibbs dynamics for the hyperbolic $Φ^{k+1}_2$-model on the plane. Our main strategy is to develop further the ideas from a recent work on the hyperbolic $Φ^3_3$-model on the three-dimensional torus by the first two authors and Okamoto (2021), and to study convergence of the so-called enhanced Gibbs measures, for which coming down from infinity for the associated SNLH with positive regularity plays a crucial role. By combining wave and heat analysis together with ideas from optimal transport theory, we then conclude global well-posedness of the hyperbolic $Φ^{k+1}_2$-model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting $Φ^{k+1}_2$-measure on the plane under the dynamics of the parabolic $Φ^{k+1}_2$-model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_03735 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hyperbolic $P(Φ)_2$-model on the plane Oh, Tadahiro Tolomeo, Leonardo Wang, Yuzhao Zheng, Guangqu Analysis of PDEs Mathematical Physics Probability 35L71, 60H15, 35K05, 35R60 We study the hyperbolic $Φ^{k+1}_2$-model on the plane. By establishing coming down from infinity for the associated stochastic nonlinear heat equation (SNLH) on the plane, we first construct a $Φ^{k+1}_2$-measure on the plane as a limit of the $Φ^{k+1}_2$-measures on large tori. We then study the canonical stochastic quantization of the $Φ^{k+1}_2$-measure on the plane thus constructed, namely, we study the defocusing stochastic damped nonlinear wave equation forced by an additive space-time white noise (= the hyperbolic $Φ^{k+1}_2$-model) on the plane. In particular, by taking a limit of the invariant Gibbs dynamics on large tori constructed by the first two authors with Gubinelli and Koch (2021), we construct invariant Gibbs dynamics for the hyperbolic $Φ^{k+1}_2$-model on the plane. Our main strategy is to develop further the ideas from a recent work on the hyperbolic $Φ^3_3$-model on the three-dimensional torus by the first two authors and Okamoto (2021), and to study convergence of the so-called enhanced Gibbs measures, for which coming down from infinity for the associated SNLH with positive regularity plays a crucial role. By combining wave and heat analysis together with ideas from optimal transport theory, we then conclude global well-posedness of the hyperbolic $Φ^{k+1}_2$-model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting $Φ^{k+1}_2$-measure on the plane under the dynamics of the parabolic $Φ^{k+1}_2$-model. |
| title | Hyperbolic $P(Φ)_2$-model on the plane |
| topic | Analysis of PDEs Mathematical Physics Probability 35L71, 60H15, 35K05, 35R60 |
| url | https://arxiv.org/abs/2211.03735 |