Regularity of the sample paths of a general second order random field

We study the sample path regularity of a second-order random field (Xt)t[set membership, variant]T where T is an open subset of . It is shown that the conditions on its covariance function, known to be equivalent to mean square differentiability of order k, imply that the sample paths are a.s. in the local Sobolev space . We discuss their necessity, and give additional conditions for the sample paths to be in a local Sobolev space of fractional order [mu]. This finally allows, via Sobolev embeddings, to draw conclusions about a.s. continuous differentiability of the sample paths.