Optimal transport has become a central tool for comparing probability measures and extracting representative distributions from heterogeneous data. Yet, in many applications the objects of interest are stochastic processes, and the classical framework ignores a key structural feature: time and information. Indeed, classical Wasserstein barycenters ignore the filtration structure, making them ill-suited for problems in mathematical finance, stochastic control, and sequential decision-making. > > We study Fréchet means with respect to the adapted Wasserstein distance, where transport plans must respect the temporal flow of information. For filtered Gaussian inputs, we establish existence and characterize when the barycenter admits an ordinary Gaussian representative via a rank criterion on a local correlation matrix. We illustrate the difference between adapted and classical barycenters through numerical experiments on autoregressive processes and briefly discuss possible applications in robust stress testing of financial models. > > Based on joint work with Madhu Gunasingam, Johannes Wiesel and Ting-Kam Leonard Wong.