We study a focusing Φ 4 3 -model with a Hartree-type nonlinearity, where the potential for the Hartree nonlinearity is given by the Bessel potential of order β. We first apply the variational argument introduced by Barashkov and Gubinelli (2018) and construct the focusing Hartree Φ 4 3 -measure for β > 2. We also show that the threshold β = 2 is sharp in the sense that the associated Gibbs measure is not normalizable for β < 2. Furthermore, we show that the following dichotomy holds at the critical value β = 2: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. We then establish a sharp almost sure global well-posedness result for the canonical stochastic quantization of the focusing Hartree Φ 4 3 -measure. Namely, we study the three-dimensional stochastic damped nonlinear wave equation (SdNLW) with a cubic nonlinearity of Hartree-type, forced by an additive space-time white noise. Using ideas from paracontrolled calculus, in particular from the recent work by Gubinelli, Koch, and the first author (2018), we prove local well-posedness of the focusing Hartree SdNLW for β > 2 (and β = 2 in the weakly nonlinear regime). In order to handle the resonant interaction, we rewrite the equation into a system of three unknowns. We then establish almost sure global well-posedness and invariance of the focusing Hartree Φ 4 3 -measure via Bourgain's invariant measure argument (1994, 1996). In view of the non-normalizability result, our almost sure global well-posedness result is sharp. In Appendix, we also consider the (parabolic) stochastic quantization for the focusing Hartree Φ 4 3 -measure and construct global-in-time invariant dynamics for β > 2 (and β = 2 in the weakly nonlinear regime).We also consider the Hartree Φ 4 3 -measure in the defocusing case. By adapting our argument from the focusing case, we first construct the defocusing Hartree Φ 4 3 -measure and the associated invariant dynamics for the defocusing Hartree SdNLW for β > 1. By introducing further renormalizations at β = 1 and β = 1 2 , we extend the construction of the defocusing Hartree Φ 4 3 -measure for β > 0, where the resulting measure is shown to be singular with respect to the reference Gaussian free field for 0 < β ≤ 1 2 .

We study the construction of the Φ 3 3 -measure and complete the program on the (non-)construction of the focusing Gibbs measures, initiated by Lebowitz, Rose, and Speer (1988). This problem turns out to be critical, exhibiting the following phase transition. In the weakly nonlinear regime, we prove normalizability of the Φ 3 3 -measure and show that it is singular with respect to the massive Gaussian free field. Moreover, we show that there exists a shifted measure with respect to which the Φ 3 3 -measure is absolutely continuous. In the strongly nonlinear regime, by further developing the machinery introduced by the authors (2020) and the first and third authors with K. Seong (2020), we establish non-normalizability of the Φ 3 3 -measure. Due to the singularity of the Φ 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, we first construct a σ-finite version of the Φ 3 3 -measure and show that this measure is not normalizable. Furthermore, we prove that the truncated Φ 3 3 -measures have no weak limit in a natural space, even up to a subsequence. We also study the dynamical problem for the canonical stochastic quantization of the Φ 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 Φ 3 3model). By adapting the paracontrolled approach, in particular from the works by Gubinelli, Koch, and the first author (2018) and by the authors (2020), we prove almost sure global well-posedness of the hyperbolic Φ 3 3 -model and invariance of the Gibbs measure in the weakly nonlinear regime. In the globalization part, we introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the Boué-Dupuis variational formula and ideas from theory of optimal transport.