The authors study the construction of the $\Phi^3_3$--measure and complete the program on the (non-)construction of the focusing Gibbs measures, initiated by Lebowitz, Rose, and Speer [J. Statist. Phys. 50 (1988), no. 3–4, 657–687]. This problem turns out to be critical, exhibiting the following phase transition. In the weakly nonlinear regime, the authors prove normalizability of the $\Phi^3_3$-measure and show that it is singular with respect to the massive Gaussian free field. Moreover, the authors show that there exists a shifted measure with respect to which the $\Phi^3_3$-measure is absolutely continuous. In the strongly nonlinear regime, by further developing the machinery tthey have introduced, , the authors establish non-normalizability of the$\Phi^3_3$-measure. Due to the singularity of the $\Phi^3_3$-measure with respect to the massive Gaussian free field, this non-normalizability part poses a particular challenge as compared to our previous works. In order to overcome this issue, the authors first construct a σ-finite version of the $\Phi^3_3$-measure and show that this measure is not normalizable. Furthermore, the authors prove that the truncated $\Phi^3_3$-measures have no weak limit in a natural space, even up to a subsequence.

The authors also study the dynamical problem for the canonical stochastic quantization of the $\Phi^3_3$measure, namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity forced by an additive space-time white noise (\(=\) the hyperbolic $\Phi^3_3$-model). By adapting the paracontrolled approach, in particular from the works by Gubinelli, Koch, and the first author [J. Eur. Math. Soc. 26 (2024), no. 3, 817–874] and by the authors [Mem. Amer. Math. Soc. 304 (2024), no. 1529], the authors prove almost sure global well-posedness of the hyperbolic $\Phi^3_3$-model and invariance of the Gibbs measure in the weakly nonlinear regime. In the globalization part, they introduce a new, conceptually simple and straightforward approach, where they directly work with the (truncated) Gibbs measure, using the Boué–Dupuis variational formula and ideas from the theory of optimal transport.