Gaussian Optimal Transport Beyond Brenier's Theorem

TL;DR

This paper advances the understanding of optimal transport for Gaussian measures in infinite-dimensional spaces, providing explicit formulas, uniqueness results, and geometric insights beyond classical theorems.

Contribution

It introduces a constructive operator-theoretic approach to characterize optimal transport maps and geodesics for Gaussian measures, extending beyond Brenier's theorem.

Findings

Optimal transport maps exist between Gaussian measures in infinite-dimensional spaces.

Explicit formulas and uniqueness of transport maps are established.

A new framework for Wasserstein barycenters using Green's operators is proposed.

Abstract

We explore the geometry of the Bures-Wasserstein space for potentially degenerate Gaussian measures on a separable Hilbert space. In this general setting, the optimal transport map is formally the subgradient of a convex function that is infinite almost everywhere, rendering conventional duality-based variational methods ineffective. We overcome this analytical barrier by exploiting a constructive operator-theoretic approach. Our central result proves that the Kantorovich problem for any pair of Gaussian measures reduces to a Monge problem; that is, an optimal transport map exists in at least one direction between two measures. This reduction allows for a complete characterization with explicit formulas for all optimal (potentially unbounded) Monge transport map and Kantorovich couplings, as well as establishing their uniqueness. Furthermore, we provide a full description of the convex set of geodesics between degenerate measures, revealing a rich geometric structure where the classical McCann interpolants arise as the extreme points. We apply these findings to construct transport maps for Gaussian processes and introduce a novel framework for Wasserstein barycenters based on random Green's operators.