We introduce a novel framework for statistical analysis of populations of nondegenerate
Gaussian processes (GPs), which are natural representations of uncertain
curves. This allows inherent variation or uncertainty in function-valued data to be
properly incorporated in the population analysis. Using the 2-Wasserstein metric we
geometrize the space of GPs with L2 mean and covariance functions over compact
index spaces. We prove uniqueness of the barycenter of a population of GPs, as well
as convergence of the metric and the barycenter of their finite-dimensional counterparts.
This justifies practical computations. Finally, we demonstrate our framework
through experimental validation on GP datasets representing brain connectivity and
climate development. A MATLAB library for relevant computations will be published
at https://sites.google.com/view/antonmallasto/software.