Focusing \Phi^4/3-model with a Hartree-type nonlinearity

Tadahiro Oh, Mamoru Okamoto, Leonardo Tolomeo

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Lebowitz, Rose, and Speer (1988) initiated the study of focusing Gibbs measures, which was continued by Brydges and Slade (1996), Bourgain (1997, 1999), and Carlen,Fröhlich, and Lebowitz (2016) among others. In this paper, we complete the program on the (non-)construction of the focusing Hartree Gibbs measures in the three-dimensional setting. More precisely, we study a focusing Φ4/3-model with a Hartree-type nonlinearity, where the potential for the Hartree nonlinearity is given by the Bessel potential of order β. We first construct the focusing Hartree Φ4/3-measure for β > 2, while we prove its nonnormalizability for β < 2. Furthermore, we establish the following phase transition at the critical value β = 2: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. We then study the canonical stochastic quantization of the focusing Hartree Φ4/3-measure, namely, the three-dimensional stochastic damped nonlinear wave equation (SdNLW) with a cubic nonlinearity of Hartree-type, forced by an additive space-time white noise, and prove almost sure global well-posedness and invariance of the
focusing Hartree Φ4/3-measure for β > 2 (and β = 2 in the weakly nonlinear regime). In view of the non-normalizability result, our almost sure global well-posedness result is sharp. In Appendix, we also discuss the (parabolic) stochastic quantization for the focusing Hartree Φ4/3-measure.
We also consider the defocusing case. By adapting our argument from the focusing case, we first construct the defocusing Hartree Φ4/3-measure and the associated invariant dynamics for the defocusing Hartree SdNLW for β > 1. By introducing further renormalizations at β = 1 and β = ½, we extend the construction of the defocusing Hartree Φ4/3-measure for β > 0,
where the resulting measure is shown to be singular with respect to the reference Gaussian free field for 0 < β ≤½.
Original languageEnglish
Number of pages126
JournalMemoirs of the american mathematical society
Publication statusAccepted/In press - 13 Jun 2022


  • Hartree \Phi^4_3-measure
  • stochastic quantization
  • stochastic nonlinear wave equation
  • nonlinear wave equation
  • Gibbs measure
  • paracontrolled calculus
  • nonlinear heat equation


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