A stochastic target approach to Ricci flow on surfaces

TL;DR

This paper introduces a novel stochastic target approach to Ricci flow on surfaces, proving exponential convergence to constant curvature metrics and developing new coupling techniques for different levels of convergence.

Contribution

It presents a new stochastic representation for Ricci flow on surfaces and introduces innovative coupling methods to analyze convergence rates, offering an alternative to traditional techniques.

Findings

Normalized Ricci flow converges exponentially to constant curvature metrics on nonpositive Euler characteristic surfaces.

Coupling of particles is used to establish convergence in various $C^k$-norms.

New stochastic and coupling techniques are developed, not previously used in Ricci flow literature.

Abstract

We develop a stochastic target representation for Ricci flow and normalized Ricci flow on smooth, compact surfaces, analogous to Soner and Touzi's representation of mean curvature flow. We prove a verification/uniqueness theorem, and then consider geometric consequences of this stochastic representation. Based on this stochastic approach, we give a proof that, for surfaces of nonpositive Euler characteristic, the normalized Ricci flow converges to a constant curvature metric exponentially quickly in every $C^k$-norm. In the case of $C^0$ and $C^1$-convergence, we achieve this by coupling two particles. To get $C^2$-convergence (in particular, convergence of the curvature), we use a coupling of three particles. This triple coupling is developed here only for the case of constant curvature metrics on surfaces, though we suspect that some variants of this idea are applicable in other situations and therefore be of independent interest. Finally, for $k\ge3$, the $C^k$-convergence follows relatively easily using induction and coupling of two particles. None of these techniques appear in the Ricci flow literature and thus provide an alternative approach to the field.