Stochastic quantization of the Φ^3_3-model

We study the construction of the Φ3 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 Φ3 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 Φ3 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 Φ3 3-measure. Due to the singularity of the Φ3 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 theΦ3 3-measure and show that this measure is not normalizable. Furthermore, we prove that the truncated Φ3 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 Φ3 3-measure, namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity forced by an additive space-time white noise (=thehyperbolicΦ3 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 Φ3 3-model and in variance 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.