A Simple Method for the Optimal Transportation

TL;DR

This paper presents a new proof for the monotonicity of Wasserstein distances between diffusions under super Ricci flow, utilizing a coupling method, and extends it to normalized L-Wasserstein distances under backward Ricci flow.

Contribution

It introduces a novel proof technique for Wasserstein distance monotonicity under Ricci flows using coupling methods, offering a new perspective in geometric analysis.

Findings

Proof of monotonicity of Wasserstein distances under super Ricci flow

Extension of the method to normalized L-Wasserstein distance under backward Ricci flow

Application of coupling method to geometric flow analysis

Abstract

In this paper we will give a new proof of the monotonicity of Wasserstein distances of two diffusions under super Ricci flow. Our proof is based on the coupling method of B.Andrew and J.Clutterbuck. The same method can also be applied to the contractivity of normalized L-Wasserstein distance under backward Ricci flow.