Brownian motion with respect to time-changing Riemannian metrics, applications to Ricci flow

TL;DR

This paper extends the concept of Brownian motion to time-dependent Riemannian metrics, especially Ricci flow, and develops related stochastic tools to analyze geometric evolution and singularities.

Contribution

It introduces a framework for Brownian motion with time-varying metrics, characterizes Ricci flow via damped parallel transport, and links stochastic analysis to geometric flow singularities.

Findings

Characterization of Ricci flow through damped parallel transport

Development of stochastic tools for time-dependent metrics

Intrinsic martingale related to flow singularities

Abstract

We generalize Brownian motion on a Riemannian manifold to the case of a family of metrics which depends on time. Such questions are natural for equations like the heat equation with respect to time dependent Laplacians (inhomogeneous diffusions). In this paper we are in particular interested in the Ricci flow which provides an intrinsic family of time dependent metrics. We give a notion of parallel transport along this Brownian motion, and establish a generalization of the Dohrn-Guerra or damped parallel transport, Bismut integration by part formulas, and gradient estimate formulas. One of our main results is a characterization of the Ricci flow in terms of the damped parallel transport. At the end of the paper we give an intrinsic definition of the damped parallel transport in terms of stochastic flows, and derive an intrinsic martingale which may provide information about singularities of the flow.