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 global-in-time dynamics of the stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing, posed on the two-dimensional torus. Our goal in this paper is two-fold. (1) By introducing a hybrid argument, combining the $I$-method in the stochastic setting with a Gronwall-type argument, we first prove global well-posedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (2) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing case. In particular, by applying Bourgain’s invariant measure argument, we prove almost sure global well-posedness of the (renormalized) defocusing SdNLW with respect to the Gibbs measure and invariance of the Gibbs measure.
We study Gibbs measures with log-correlated base Gaussian fields on the ddimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson's argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove non-normalizability of the Gibbs measure. When d = 2, our argument provides an alternative proof of the non-normalizability result for the focusing Φ 4 2 -measure by Brydges and Slade (1996). We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) non-normalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.
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