Focusing $\Phi^4_3$-model with a Hartree-type nonlinearity

Abstract

(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) 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, Fr\"ohlich, 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 $\Phi^4_3$-model with a Hartree-type nonlinearity, where the potential for the Hartree nonlinearity is given by the Bessel potential of order $\beta$. We first construct the focusing Hartree $\Phi^4_3$-measure for $\beta > 2$, while we prove its non-normalizability for $\beta < 2$. Furthermore, we establish the following phase transition at the critical value $\beta = 2$: 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 $\Phi^4_3$-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 $\Phi^4_3$-measure for $\beta > 2$ (and $\beta = 2$ 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 $\Phi^4_3$-measure. We also construct the defocusing Hartree $\Phi^4_3$-measure for $\beta > 0$.