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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , a Brownian motion depending on the deformation of the manifold (via the family <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> and a characterization of the Ricci flow in terms of the damped parallel transport.