The geometry of the adapted Bures--Wasserstein space

TL;DR

This paper develops the geometric structure of the adapted Bures--Wasserstein space for Gaussian processes, revealing it as an Alexandrov space with non-negative curvature and detailed tangent and exponential map descriptions.

Contribution

It introduces the first geometric theory for the adapted Bures--Wasserstein space, including curvature, tangent cones, and the structure of Gaussian process subspaces.

Findings

The space is an Alexandrov space with non-negative curvature.

Explicit descriptions of tangent cones and exponential maps are provided.

Gaussian processes form a geodesically convex subspace with linear tangent cones.

Abstract

The adapted Bures--Wasserstein space consists of Gaussian processes endowed with the adapted Wasserstein distance. It can be viewed as the analogue of the classical Bures--Wasserstein space in optimal transport for the setting of stochastic processes, where the standard Wasserstein distance is inadequate and has to be replaced by its adapted counterpart. We develop a comprehensive geometric theory for the adapted Bures--Wasserstein space, thereby also providing the first results on the fine geometric structure of adapted optimal transport. In particular, we show that the adapted Bures--Wasserstein space is an Alexandrov space with non-negative curvature and provide explicit descriptions of tangent cones and exponential maps. Moreover, we show that Gaussian processes satisfying a natural non-degeneracy condition form a geodesically convex subspace. This subspace is characterized precisely by the property that its tangent cones are linear and hence coincide with the tangent space.