Hyperbolic P ( Φ ) 2 -model on the Plane

In this paper, we construct invariant Gibbs dynamics for the hyperbolic ${\Phi }_2^{k+1}$ -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 ${\Phi }_2^{k+1}$ -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 ${\Phi }_2^{k+1}$ -measure on the plane as a limit of the ${\Phi }_2^{k+1}$ -measures on large tori. (ii) We then construct invariant Gibbs dynamics for the hyperbolic ${\Phi }_2^{k+1}$ -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 ${\Phi }_3^3$ -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 ${\Phi }_2^{k+1}$ -model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting ${\Phi }_2^{k+1}$ -measure on the plane under the dynamics of the parabolic ${\Phi }_2^{k+1}$ -model.