Brownian motion with respect to a metric depending on time; definition, existence and applications to Ricci flow

Abstract

Given an n-dimensional compact manifold M, endowed with a family of Riemannian metrics g ( t ) , a Brownian motion depending on the deformation of the manifold (via the family g ( t ) of metrics) is defined. This tool enables a probabilistic view of certain geometric flows (e.g. Ricci flow, mean curvature flow). In particular, we give a martingale representation formula for a non-linear PDE over M, as well as a Bismut type formula for a geometric quantity which evolves under this flow. As application we present a gradient control formula for the heat equation over ( M , g ( t ) ) and a characterization of the Ricci flow in terms of the damped parallel transport. To cite this article: M. Arnaudon et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).