Invariance of $\phi^4$ measure under nonlinear wave and Schr\"odinger equations on the plane

We show almost sure wellposedness of mild solution to the cubic nonlinear wave equation in a weighted Besov space over $\mathbb R^2$. To achieve this, we show that any weak limit of $\phi^4$ measures on increasing tori is invariant under the equation. We review and slightly simplify the periodic theory and the construction of the weak limit measure, and then use finite speed of propagation to reduce the infinite-volume case to the previous setup. Our argument also gives a weaker invariance result on the nonlinear Schr\"odinger equation in the same setting.