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Hyperbolic $P(Φ)_2$-model on the plane

Tadahiro Oh, Leonardo Tolomeo, Yuzhao Wang, Guangqu Zheng

TL;DR

The paper advances the stochastic quantization of the two-dimensional hyperbolic $ ext{Φ}^{k+1}_2$-model on the plane by constructing a plane Gibbs measure as a large-torus limit and establishing global well-posedness and invariance for SdNLW with Gibbs-distributed data. It develops a rigorous framework of enhanced Gibbs measures to control the interplay between wave and heat analyses, leverages coming-down-from-infinity for the stochastic heat equation, and transfers invariant Gibbs dynamics from large tori to $bR^2$. A key technical achievement is the convergence of enhanced data sets and the associated Gibbs measures in weighted function spaces, ensuring invariance and global dynamics on the plane. The results also yield invariance of the parabolic $ ext{Φ}^{k+1}_2$-measure under the corresponding parabolic dynamics, linking parabolic and hyperbolic stochastic quantization in an unbounded setting.

Abstract

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.

Hyperbolic $P(Φ)_2$-model on the plane

TL;DR

The paper advances the stochastic quantization of the two-dimensional hyperbolic -model on the plane by constructing a plane Gibbs measure as a large-torus limit and establishing global well-posedness and invariance for SdNLW with Gibbs-distributed data. It develops a rigorous framework of enhanced Gibbs measures to control the interplay between wave and heat analyses, leverages coming-down-from-infinity for the stochastic heat equation, and transfers invariant Gibbs dynamics from large tori to . A key technical achievement is the convergence of enhanced data sets and the associated Gibbs measures in weighted function spaces, ensuring invariance and global dynamics on the plane. The results also yield invariance of the parabolic -measure under the corresponding parabolic dynamics, linking parabolic and hyperbolic stochastic quantization in an unbounded setting.

Abstract

We study the hyperbolic -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 -measure on the plane as a limit of the -measures on large tori. We then study the canonical stochastic quantization of the -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 -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 -model on the plane. Our main strategy is to develop further the ideas from a recent work on the hyperbolic -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 -model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting -measure on the plane under the dynamics of the parabolic -model.
Paper Structure (22 sections, 26 theorems, 439 equations)

This paper contains 22 sections, 26 theorems, 439 equations.

Table of Contents

  1. Introduction
  2. Hyperbolic $\Phi^{k+1}_2$-model
  3. Review of the periodic problem
  4. Main result
  5. Preliminary
  6. Notations
  7. Weighted Sobolev and Besov spaces
  8. Linear estimates on weighted Sobolev spaces
  9. Product estimates
  10. Tools from stochastic analysis
  11. Coming down from infinity
  12. Coming down from infinity on weighted Lebesgue spaces
  13. Coming down from infinity on weighted Sobolev spaces of positive regularities
  14. Construction of the Gibbs measure $\vec{\rho}_\infty$ on the plane
  15. Global well-posedness and invariance of the Gibbs measure
  16. ...and 7 more sections

Key Result

Theorem 1.2

Let $k \in 2\mathbb{N}+1$. There exists a subset $\mathcal{A} = \{ L_j: j \in \mathbb{N}\} \subset \mathbb{N}$(with $L_j < L_{j'}$ for $j < j'$) such that the following statements hold. (i) Let $s < 0$, finite $p \ge 1$, and $\mu > 0$. Let $\vec{\rho}_L$ be the Gibbs measure on the dilated torus $\m where $\rho_\infty$ is the $\Phi^{k+1}_2$-measure on $\mathbb{R}^2$ constructed as a limit of the $

Theorems & Definitions (59)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark 2.4
  • ...and 49 more