  We consider the optimal transport problem between zero mean Gaussian
stationary random fields both in the aperiodic and periodic case. We show that
the solution corresponds to a weighted Hellinger distance between the
multivariate and multidimensional power spectral densities of the random
fields. Then, we show that such a distance defines a geodesic, which depends on
the weight function, on the manifold of the multivariate and multidimensional
power spectral densities.
