刘维尔共形域论的随机量子化
Stochastic quantization of Liouville conformal field theory
摘要：
研究了闭曲面上受时空白噪声强迫的非线性随机热方程，方程的非线性项为$e^{βu}$。这个方程对应于刘维尔量子引力（LQG）测度的随机量子化。(i)我们首先回顾了$L^2$体系$0<β<\sqrt{2}$中的刘维共形域论（LCFT）中LQG测度的构造。这统一了David-Kupiainen-Rhodes-Vargas（2016），David-Rhodes-Vargas（2016）和Guillarmou-Rhodes-Vargas（2019）的方法，这些方法分别处理亏格为0，1和$> 1$的闭曲面的情况。此外，我们的论点表明，这一措施是独立的近似程序的一大类光滑近似。(ii)我们证明了抛物型随机动力学的几乎必然的整体适定性，以及测度在这种随机流下的不变性。特别地，我们的结果改进了Garban（2020）在球和环面的正则度量情况下的先前结果，并且在具有更高亏格的闭曲面的情况下是新的。
英文摘要：
We study a nonlinear stochastic heat equation forced by a space-time white noise on closed surfaces, with nonlinearity $e^{βu}$. This equation corresponds to the stochastic quantization of the Liouville quantum gravity (LQG) measure. (i) We first revisit the construction of the LQG measure in Liouville conformal field theory (LCFT) in the $L^2$ regime $0<β<\sqrt{2}$. This uniformizes in this regime the approaches of David-Kupiainen-Rhodes-Vargas (2016), David-Rhodes-Vargas (2016) and Guillarmou-Rhodes-Vargas (2019) which treated the case of a closed surface with genus 0, 1 and $> 1$ respectively. Moreover, our argument shows that this measure is independent of the approximation procedure for a large class of smooth approximations. (ii) We prove almost sure global well-posedness of the parabolic stochastic dynamics, and invariance of the measure under this stochastic flow. In particular, our results improve previous results obtained by Garban (2020) in the cases of the sphere and the torus with their canonical metric, and are new in the case of closed surfaces with higher genus.
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