  We are interested in the regularity of centered Gaussian processes (Z_x), x
in M indexed by compact metric spaces M. 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_xZ_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.
