Differentiable maps between Wasserstein spaces

TL;DR

This paper introduces a global notion of differentiability for maps between Wasserstein spaces of Riemannian manifolds, focusing on pushforward maps and convex combinations, with an explicit differential construction.

Contribution

It proposes the first global differentiability framework for Wasserstein space maps, extending classical notions without pointwise assumptions.

Findings

Defined a global differentiability concept for Wasserstein maps

Analyzed differentiability of pushforward maps induced by smooth manifold maps

Constructed explicit differentials for convex combinations of differentiable maps

Abstract

A notion of differentiability is being proposed for maps between Wasserstein spaces of order 2 of smooth, connected and complete Riemannian manifolds. Due to the nature of the tangent space construction on Wasserstein spaces, we only give a global definition of differentiability, i.e. without a prior notion of pointwise differentiability. With our definition, however, we recover the expected properties of a differential. Special focus is being put on differentiability properties of pushforward maps induced by smooth maps between the underlying manifolds, and on convex mixing of differentiable maps, with an explicit construction of the differential.