In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville conformal field theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus, in the full “$L^1$ regime” $\sigma < \sigma _{L^1}=2 \sqrt \pi $ where $\sigma $ is the noise strength. We also describe the main necessary modifications needed for the SRF on general compact surfaces and list some open questions.

中文翻译：

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紧致表面上的随机 Ricci 流

在本文中，我们在两个空间维度上介绍了随机 Ricci 流 (SRF)。流动相对于由刘维尔共形场理论引起的测量是对称的。使用狄利克雷形式的理论，我们构造了平面环面上面积测度相关方程的弱解，在完整的“$L^1$ 状态”$\sigma < \sigma _{L^1}=2 \sqrt \pi $ 其中$\sigma $ 是噪声强度。我们还描述了 SRF 在一般紧凑表面上所需的主要必要修改，并列出了一些未解决的问题。