NSTITUTE 2015 : 06 Regularity of Gaussian Processes on Dirichlet Spaces

We are interested in the regularity of centered Gaussian processes (Z x (ω)) x∈M indexed by compact metric spaces (M, ρ). It is shown that the almost everywhere Besov space regularity of such a process is (almost) equivalent to the Besov regularity of the covariance K(x, y) = E(Z x Z y) under the assumption that (i) there is an underlying Dirichlet structure on M which determines the Besov space regularity, and (ii) the operator K with kernel K(x, y) and the underlying operator A of the Dirichlet structure commute. As an application of this result we establish the Besov regularity of Gaussian processes indexed by compact homogeneous spaces and, in particular, by the sphere.