On Parallel Transport in Wasserstein Space

TL;DR

This paper presents a construction of parallel transport within Wasserstein space, providing a complete proof for the linear tangent space and establishing equivalence with existing methods.

Contribution

It introduces a new construction for parallel transport in Wasserstein space and proves its validity, connecting it with prior approaches.

Findings

Complete proof for linear tangent space parallel transport

Construction for full tangent cones based on natural lemmas

Equivalence with existing literature methods

Abstract

In this short note, we would like to give a construction of parallel transport for tangent cones lying in the interior of a geodesic in Wasserstein space. We give a complete proof for the linear part of the tangent space, and show that a construction for the full tangent cones follows from some natural lemmas on Wasserstein space. It can easily be shown that our construction is equivalent to those used in the previous literature on this subject.