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We show probabilistic existence and uniqueness for the Wick-ordered cubicnonlinear wave equation in a weighted Besov space over $\mathbb R^2$. Toachieve this, we show that a weak limit of $\phi^4$ measures on increasing toriis invariant under the equation. We review and slightly simplify the periodictheory and the construction of the weak limit measure, and then use finitespeed of propagation to reduce the infinite-volume case to the previous setup.Our argument also gives a weak (Albeverio--Cruzeiro) invariance result on thenonlinear Schr\"odinger equation in the same setting.

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