论文标题
聚焦$φ^4_3 $ -MODEL与Hartree型非线性
Focusing $Φ^4_3$-model with a Hartree-type nonlinearity
论文作者
论文摘要
（由于Arxiv设置的摘要的字符数量的限制，无法在此处显示完整的摘要。请参阅本文中的摘要。）Lebowitz，Rose和Speer（1988）开始研究Focus Gobosing Gibbs测量的研究，Brydges and Slade（1996）继续进行，该研究由Brydges and Slade（1996），其他Bourgain（1997，1999，1999，1999），和Carlen，Frichen，以及Carlens，frich and＆frich，以及Lehlich and h。在本文中，我们完成了有关在三维环境中（非）构造哈特里·吉布斯（Hartree Gibbs）措施的（非）构造的计划。更确切地说，我们研究了一个具有hartree型非线性的聚焦$φ^4_3 $ - 模型，其中bessel的贝塞尔潜力$β$给出了Hartree非线性的潜力。我们首先构建了焦点hartree $φ^4_3 $ - $β> 2 $的量化，而我们证明其对$β<2 $的非正相关性。此外，我们在临界值$β= 2 $上建立以下相变：在弱非线性状态中的正常化性和强烈非线性方案中的非差异性。然后，我们研究了聚焦hartree $φ^4_3 $的规范随机量化，即三维随机抑制的非线性波动方程（SDNLW），其与Hartree型的立方非线性相比，由Hartree型的非线性，几乎是由增添空间白色噪声和pocute poction and Poction and Pocuts and poction and poction and poction and poction and poction and poction and pocutiance and Invarcience for and Invarciance and Invarciance for new offiance $φ^4_3 $ - $β> 2 $（和弱非线性制度中的$β= 2 $）。鉴于不符合性的结果，我们几乎确定的全球适应性结果是敏锐的。在附录中，我们还讨论了焦点hartree $φ^4_3 $ measure的（抛物线）随机量化。 我们还构建了deDocusing hartree $φ^4_3 $ - $β> 0 $的限制。
(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 three-dimensional setting. More precisely, 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 construct the focusing Hartree $Φ^4_3$-measure for $β> 2$, while we prove its non-normalizability 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 $Φ^4_3$-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 $Φ^4_3$-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 $Φ^4_3$-measure. We also construct the defocusing Hartree $Φ^4_3$-measure for $β> 0$.
