We develop statistical models for samples of distribution-valued stochastic processes featuring time-indexed univariate distributions, with emphasis on functional principal component analysis. The proposed model presents an intrinsic rather than transformation-based approach. The starting point is a transport process representation for distribution-valued processes under the Wasserstein metric. Substituting transports for distributions addresses the challenge of centering distribution-valued processes and leads to a useful and interpretable decomposition of each realized process into a process-specific single transport and a real-valued trajectory. This representation makes it possible to utilize a scalar multiplication operation for transports and facilitates not only functional principal component analysis but also to introduce a latent Gaussian process. This Gaussian process proves especially useful for the case where the distribution-valued processes are only observed on a sparse grid of time points, establishing an approach for longitudinal distribution-valued data. We study the convergence of the key components of this novel representation to their population targets and demonstrate the practical utility of the proposed approach through simulations and several data illustrations.

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