Coupling of Brownian motions and Perelman's L-functional

TL;DR

This paper demonstrates that on a manifold evolving under backwards Ricci flow, two Brownian motions can be coupled to ensure the expected normalized L-distance does not increase, providing new insights into heat equation solutions.

Contribution

It introduces a novel coupling method for Brownian motions under Ricci flow, leading to a new proof of non-increasing normalized L-transportation cost between heat equation solutions.

Findings

Coupling of Brownian motions with non-increasing expected L-distance

New proof of non-increasing normalized L-transportation cost

Application to heat equation solutions under Ricci flow

Abstract

We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that the expectation of their normalized L-distance is non-increasing. As an immediate corollary we obtain a new proof of a recent result of Topping (J. reine angew. Math. 636 (2009), 93-122), namely that the normalized L-transportation cost between two solutions of the heat equation is non-increasing as well.