# Hyperbolic P(Φ)2-model on the Plane

## TL;DR

The paper develops a mathematical framework for constructing invariant dynamics in a specific nonlinear wave model on the plane.

## Contribution

A novel approach to constructing invariant Gibbs dynamics for the hyperbolic Φ²-model on the plane using enhanced Gibbs measures and convergence techniques.

## Key findings

- Invariant Gibbs dynamics for the hyperbolic Φ²-model on the plane are constructed as a limit of dynamics on large tori.
- The limiting Φ²-measure on the plane is shown to be invariant under both hyperbolic and parabolic dynamics.
- Global well-posedness of the hyperbolic Φ²-model on the plane is established.

## Abstract

In this paper, we construct invariant Gibbs dynamics for the hyperbolic \documentclass[12pt]{minimal}
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\begin{document}$$\Phi ^{k+1}_2$$\end{document}Φ2k+1-model (namely, defocusing stochastic damped nonlinear wave equation forced by an additive space-time white noise) on the plane. (i) For this purpose, we first revisit the construction of a \documentclass[12pt]{minimal}
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\begin{document}$$\Phi ^{k+1}_2$$\end{document}Φ2k+1-measure on the plane. More precisely, by establishing coming down from infinity for the associated stochastic nonlinear heat equation (SNLH) on the plane, we first construct a \documentclass[12pt]{minimal}
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\begin{document}$$\Phi ^{k+1}_2$$\end{document}Φ2k+1-measure on the plane as a limit of the \documentclass[12pt]{minimal}
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\begin{document}$$\Phi ^{k+1}_2$$\end{document}Φ2k+1-measures on large tori. (ii) We then construct invariant Gibbs dynamics for the hyperbolic \documentclass[12pt]{minimal}
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\begin{document}$$\Phi ^{k+1}_2$$\end{document}Φ2k+1-model on the plane, by taking a limit of the invariant Gibbs dynamics on large tori constructed by the first two authors with Gubinelli and Koch (Int Math Res Not 21:16954–16999,2022). Here, our main strategy is to develop further the ideas from a recent work on the hyperbolic \documentclass[12pt]{minimal}
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\begin{document}$$\Phi ^3_3$$\end{document}Φ33-model on the three-dimensional torus by the first two authors and Okamoto (Mem Eur Math Soc 16, 2025), 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 \documentclass[12pt]{minimal}
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\begin{document}$$\Phi ^{k+1}_2$$\end{document}Φ2k+1-model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting \documentclass[12pt]{minimal}
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\begin{document}$$\Phi ^{k+1}_2$$\end{document}Φ2k+1-measure on the plane under the dynamics of the parabolic \documentclass[12pt]{minimal}
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\begin{document}$$\Phi ^{k+1}_2$$\end{document}Φ2k+1-model.