The tangent space to the Wasserstein space: parallel transport and other applications

TL;DR

This paper introduces a new formal tangent space to the Wasserstein space, enabling the definition of parallel transport, regularity, and curve translation within this geometric framework.

Contribution

It proposes a novel notion of tangent space to Wasserstein space that generalizes previous concepts and facilitates advanced geometric operations.

Findings

Defined a new tangent space concept for Wasserstein space

Enabled the formulation of parallel transport in Wasserstein space

Established notions of regularity and translation of curves

Abstract

We propose a new notion of the formal tangent space to the Wasserstein space $\mathcal{P}(X)$ at a given measure. Modulo an integrability condition, we say that this tangent space is made of functions over $X$ which are valued in the probability measures over the tangent bundle to $X$. This generalization of previous concepts of tangent spaces allows us to define appropriate notions of parallel transport, $\mathcal{C}^{1,\alpha}$ regularity over $\mathcal{P}(X)$ and translation of a curve over $\mathcal{P}(X)$.