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We study the nonlinear wave equation (NLW) on the two-dimensional torus$\mathbb T^2$ with Gaussian random initial data on $H^s(\mathbb T^2) \timesH^{s-1}(\mathbb T^2)$, $s < 0$, distributed according to the base Gaussian freefield $\mu$ associated with the invariant Gibbs measure studied by Thomann andthe first author (2020). In particular, we investigate the approximationproperty of the corresponding solution by smooth (random) solutions. Our mainresults in this paper are two-fold. (i) We show that the solution map for therenormalized cubic NLW defined on the Gaussian free field $\mu$ is the uniqueextension of the solution map defined for smoothed Gaussian initial dataobtained by mollification, independent of mollification kernels. (ii) We alsoshow that there is a regularization of the Gaussian initial data so that thecorresponding smooth solutions almost surely have no limit in the naturaltopology. This second result in particular states that one can not usearbitrary smooth approximation for the renormalized cubic NLW dynamics.  As a preliminary step for proving (ii), we establish a (deterministic) norminflation result at general initial data for the (unrenormalized) cubic NLW on$\mathbb T^d$ and $\mathbb R^d$ in negative Sobolev spaces, extending the norminflation result by Christ, Colliander, and Tao (2003).

Uniqueness and non-uniqueness of the Gaussian free field evolution under the two-dimensional Wick ordered cubic wave equation