Regularity of Gaussian Processes on Dirichlet Spaces

Abstract

We study the regularity of centered Gaussian processes $$(Z_x( \omega ))_{x\in M}$$ , indexed by compact metric spaces $$(M, \rho )$$ . It is shown that the almost everywhere Besov regularity of such a process is (almost) equivalent to the Besov regularity of the covariance $$K(x,y) = {\mathbb E}(Z_x Z_y)$$ under the assumption that (i) there is an underlying Dirichlet structure on M that determines the Besov 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.