Stochastic Ricci Flow on Compact Surfaces

2019/04/24 by Julien Dubédat, Hao Shen, Dubédat, Julien +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1904.10909

openalex publication_date 2019/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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 "L¹ regime" σ< σ_L¹=2√π where σ is the noise strength. We also describe the main necessary modifications needed for the SRF on general compact surfaces, and list some open questions.

Citations

• Equivalence of Liouville measure and Gaussian free field
• Harmonic Mappings of Riemannian Manifolds
• The motion of a random string
• Constructive Liouville Conformal Field Theory
• An analytic BPHZ theorem for regularity structures
• The dynamical sine-Gordon model in the full subcritical regime
• On the parabolic and hyperbolic Liouville equations

Cited by

• Stochastic conformal flows in even dimensions

Related

• Ricci Flow in Two Dimensions
• Two-dimensional Ricci flow as a stochastic process
• Stochastic conformal flows in even dimensions
• Stochastic quantization of Liouville conformal field theory
• Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces