Gaussianity of adapted Wasserstein barycenters for Gaussian marginals

Establish the existence of an adapted Wasserstein barycenter with respect to the adapted 2-Wasserstein distance for a collection of Gaussian distributions on R^N and prove that at least one such barycenter is itself a Gaussian distribution. Concretely, given positive weights summing to one and Gaussian measures {μ_i} on R^N, determine whether the optimization problem inf_{ν in P_2(R^N)} ∑_i w_i AW_2^2(μ_i, ν) admits a minimizer and show that there exists a Gaussian minimizer ν*.

Background

The paper derives an explicit formula for the adapted 2-Wasserstein distance AW_2 between non-degenerate Gaussian processes in discrete time and characterizes optimal bicausal couplings, introducing an adapted Bures–Wasserstein distance on covariance matrices. This provides foundational tools for studying adapted optimal transport among Gaussian laws.

In classical Wasserstein geometry, barycenters with respect to the W_2 metric are known to exist under mild conditions, and when the input measures are Gaussian, the Wasserstein barycenter is Gaussian (see Agueh and Carlier, 2011). The adapted Wasserstein barycenter was introduced recently (AKP24), but its structure in the Gaussian setting is not established. The authors explicitly conjecture the existence of an adapted Wasserstein barycenter that remains within the Gaussian family when the marginals are Gaussian, mirroring the classical W_2 result.