Optimal transport between gaussian random fields

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.