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∈T(Xt)t∈T where TT is an open subset of RdRd. It is shown that the conditions on its covariance function, known to be equivalent to mean square differentiability of order kk, imply that the sample paths are a.s. in the local Sobolev space Wlock,2(T). We discuss their necessity, and give additional conditions for the sample paths to be in a local Sobolev space Wlocμ,2(T) of fractional order μμ. This finally allows, via Sobolev embeddings, to draw conclusions about a.s. continuous differentiability of the sample paths.