Hyperbolic $P(\Phi)_2$-model on the plane

We study the hyperbolic $\Phi^{k+1}_2$-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 $\Phi^{k+1}_2$-measure on the plane as a limit of the $\Phi^{k+1}_2$-measures on large tori. We then study the canonical stochastic quantization of the $\Phi^{k+1}_2$-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 $\Phi^{k+1}_2$-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 $\Phi^{k+1}_2$-model on the plane. 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 (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 $\Phi^{k+1}_2$-model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting $\Phi^{k+1}_2$-measure on the plane under the dynamics of the parabolic $\Phi^{k+1}_2$-model.