Focusing $\Phi^4_3$model with a Hartreetype 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, 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 threedimensional setting. More precisely, we study a focusing $\Phi^4_3$model with a Hartreetype 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 nonnormalizability for $\beta < 2$. Furthermore, we establish the following phase transition at the critical value $\beta = 2$: normalizability in the weakly nonlinear regime and nonnormalizability in the strongly nonlinear regime. We then study the canonical stochastic quantization of the focusing Hartree $\Phi^4_3$measure, namely, the threedimensional stochastic damped nonlinear wave equation (SdNLW) with a cubic nonlinearity of Hartreetype, forced by an additive spacetime white noise, and prove almost sure global wellposedness 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 nonnormalizability result, our almost sure global wellposedness 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 $\beta > 0$.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 DOI:
 10.48550/arXiv.2009.03251
 arXiv:
 arXiv:2009.03251
 Bibcode:
 2020arXiv200903251O
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 35L71;
 60H15;
 81T08;
 35K15
 EPrint:
 126 pages. Minor modifications. To appear in Mem. Amer. Math. Soc