Entropic optimal transport beyond product reference couplings: the Gaussian case on Euclidean space

The Optimal Transport (OT) problem with squared Euclidean cost consists in finding a coupling between two input measures that maximizes correlation. Consequently, the optimal coupling is often singular with respect to the Lebesgue measure. Regularizing the OT problem with an entropy term yields an approximation called entropic optimal transport. Entropic penalties steer the induced coupling toward a reference measure with desired properties. For instance, when seeking a diffuse coupling, the most popular reference measures are the Lebesgue measure and the product of the two input measures. In this work, we study the case where the reference coupling is not a product, focussing on the Gaussian case as a core paradigm. We establish a reduction of such a regularised OT problem to a matrix optimization problem, enabling us to provide a complete description of the solution, both in terms of the primal variable and the dual variables. Beyond its intrinsic interest, allowing non-product references is essential in dynamic statistical settings. As a key motivation, we address the reconstruction of trajectory dynamics from finitely many time marginals where, unlike product references, Gaussian process references produce transitions that assemble into a coherent continuous-time process.