  We show probabilistic existence and uniqueness for the Wick-ordered cubic
nonlinear wave equation in a weighted Besov space over $\mathbb R^2$. To
achieve this, we show that a 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 weak (Albeverio--Cruzeiro) invariance result on the
nonlinear Schr\"odinger equation in the same setting.
