The existence of geodesics in Wasserstein spaces over path groups and loop groups

TL;DR

This paper proves the existence and uniqueness of optimal transport maps in Wasserstein spaces over path and loop groups, providing explicit solutions for certain cases and demonstrating the existence of geodesics between probability measures.

Contribution

It establishes the existence and uniqueness of optimal transport maps in Wasserstein spaces over path and loop groups, including explicit formulas for specific cases.

Findings

Existence and uniqueness of optimal transport maps for p>1

Explicit expression of the optimal transport map for p=2

Existence of geodesics connecting probability measures on path and loop groups

Abstract

In this work we prove the existence and uniqueness of the optimal transport map for $L^p$-Wasserstein distance with $p>1$, and particularly present an explicit expression of the optimal transport map for the case $p=2$. As an application, we show the existence of geodesics connecting probability measures satisfying suitable condition on path groups and loop groups.