Optimal Transport between Gaussian random fields

Abstract: 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.