Stochastic Ricci Flow on Surfaces

Explore the concept of Stochastic Ricci flow in two spatial dimensions through this 50-minute lecture from the Hausdorff Junior Trimester Program on Randomness, PDEs, and Nonlinear Fluctuations. Delve into the formal representation of the flow using evolving Riemannian metrics with space-time noise, and examine its formulation in terms of the conformal factor as a quasi-linear generalization of the stochastic heat equation. Discover the flow's symmetry with respect to a measure induced by Liouville Conformal Field Theory. Learn about the construction of weak solutions for the associated equation of the area measure on a flat torus using Dirichlet forms theory. Understand the necessary modifications for applying Stochastic Ricci flow to general compact surfaces due to conformal anomaly. Investigate the behavior of the total surface area under the flow, which follows a squared Bessel process. Conclude by discussing open questions in this field of study.