The geometry of the adapted Bures--Wasserstein space

TL;DR

This work develops a comprehensive geometric theory for the adapted Bures--Wasserstein space of Gaussian processes under the adapted Wasserstein distance ${\mathcal{AW}}_2$. It proves that the ABW space, realized as the quotient $({\bf L}/_{\bf O}, d_{\rm ABW})$, is an Alexandrov space with non-negative curvature and provides explicit Procrustes-based descriptions of tangent cones and exponential maps; it also identifies a geodesically convex subspace ${\bf L}^{\rm reg}/_{\bf O}$ with linear tangent cones and tractable geodesic geometry. The paper then extends these insights to the broader adapted transport setting, showing that the Wasserstein space of filtered processes $({\rm FP}_2, {\mathcal{AW}}_2)$ is non-negatively curved and that Gaussian processes form an Alexandrov space with explicit AW_2-geodesics given by $\gamma(u)={[((1-u)L+uMP)G]}$ for $P\in {\bf O}^*(L,M)$. It also provides a precise AW_2 distance formula for Gaussian processes, studies the geometry of geodesics, angles, and tangent cones via the Procrustes optimizers ${\bf O}^*(L,M)$, and characterizes regular points where tangent cones are linear. Collectively, these results lay the foundation for a robust, Riemannian-like analysis of adapted optimal transport in stochastic-process settings and illuminate how filtration structure shapes geometry and interpolation between Gaussian processes.

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.

Theorems & Definitions (74)

• Remark 1.1
• Remark 1.2
• Definition 1.3
• Theorem 1.4
• Remark 1.5
• Example 1.6
• Theorem 1.7
• Proposition 1.8
• Theorem 1.9
• Corollary 1.10