  Monge-Kantorovich distances, otherwise known as Wasserstein distances, have
received a growing attention in statistics and machine learning as a powerful
discrepancy measure for probability distributions. In this paper, we focus on
forecasting a Gaussian process indexed by probability distributions. For this,
we provide a family of positive definite kernels built using transportation
based distances. We provide a probabilistic understanding of these kernels and
characterize the corresponding stochastic processes. We prove that the Gaussian
processes indexed by distributions corresponding to these kernels can be
efficiently forecast, opening new perspectives in Gaussian process modeling.
