I will discuss well-posedness of the stochastic damped nonlinear wave equation (SdNLW) forced by a space-time white noise with the Gibbsian initial data. This problem is also known as the hyperbolic $\Phi^{k+1} _2$-model since it corresponds to the so-called canonical stochastic quantization of the $\Phi^{k+1} _2$-measure. In this talk, our main goal is to study this problem on the plane.  Previously, this problem was studied on the two-dimensional torus by Gubinelli-Koch-Oh-Tolomeo (2021). By introducing a proper renormalization, they constructed global-in-time invariant Gibbs dynamics. By taking a large torus limit, I aim to construct invariant Gibbs dynamics for the hyperbolic $\Phi^{k+1} _2$-model on the plane.  First, I plan to go over the construction of a $\Phi^{k+1} _2$-measure on the plane as a limit of the $\Phi^{k+1} _2$-measures on large tori. This is done by establishing coming-down-from-infinity for the associated stochastic nonlinear heat equation (SNLH) on the plane. We then construct invariant Gibbs dynamics for the hyperbolic $\Phi^{k+1} _2$-model on the plane by taking a limit of the invariant Gibbs dynamics on large tori. Our strategy is inspired by a recent work by Oh-Okamoto-Tolomeo (2021) on the hyperbolic $\Phi^3 _3$-model on the three-dimensional torus, where we reduce the problem to studying convergence of the so-called extended Gibbs measures. By combining wave and heat analysis together with ideas from optimal transport theory, I establish convergence of the extended Gibbs measures.

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