Stochastic Quantization of the 𝛟³₃-Model
Tadahiro Oh, Mamoru Okamoto, Leonardo Tolomeo · Memoirs of the European Mathematical Society · 2025
We study the construction of the ˆ3 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, we prove normalizability of the ˆ3 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 ˆ3 3 -measure is absolutely continuous.In the strongly nonlinear regime, by further developing the machinery introduced by the authors, we establish non-normalizability of the ˆ3 3 -measure.Due to the singularity of the ˆ3 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 -finite version of the ˆ3 3 -measure and show that this measure is not normalizable.Furthermore, we prove that the truncated ˆ3 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 ˆ3 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 ˆ3 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], we prove almost sure global well-posedness of the hyperbolic ˆ3 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.