A Characterization of Gaussian Processes over Flat Tori

Gaussian processes provide a flexible probabilistic framework for modeling functions defined on structured domains. When the domain is a Riemannian manifold, covariance kernels can be constructed via the spectral decomposition of the Laplace-Beltrami operator, leading in particular to Matern kernels through SPDE formulations. While this approach is well understood for the standard flat torus, less explicit characterizations are available for general flat tori associated with arbitrary lattices. In this work, we provide a simple and computationally convenient description of Gaussian processes on flat tori by explicitly characterizing the spectrum and eigenfunctions of the Laplacian in terms of the dual lattice. This yields closed-form expressions for Matern kernels on general flat tori that closely parallel the standard torus case.