Stochastic Ricci Flow on Compact Surfaces

Abstract

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.

arXiv comment42 pages

References in corpus (15)

• Liouville Field Theory -- A decade after the revolution
• Renormalising SPDEs in regularity structures
• An analytic BPHZ theorem for regularity structures
• The dynamical sine-Gordon model
• Liouville Quantum Gravity on the complex tori
• Langevin dynamic for the 2D Yang-Mills measure
• Geometric stochastic heat equations
• A solution theory for quasilinear singular SPDEs
• Liouville heat kernel: regularity and bounds
• The motion of a random string
• The dynamical sine-Gordon model in the full subcritical regime
• The Ricci flow on the sphere with marked points
• Equivalence of Liouville measure and Gaussian free field
• Constructive Liouville Conformal Field Theory
• On the parabolic and hyperbolic Liouville equations

Cited by in corpus (4)

• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• Stochastic quantization associated with the $\exp (\Phi {)}_2$-quantum field model driven by space-time white noise on the torus
• Stochastic quantization of Liouville conformal field theory
• On the parabolic and hyperbolic Liouville equations