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

We study the sample path regularity of a second-order random field ( X t ) t ∈ T where T is an open subset of R d . 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 W loc k , 2 ( T ) . We discuss their necessity, and give additional conditions for the sample paths to be in a local Sobolev space W loc μ , 2 ( T ) of fractional order μ . This finally allows, via Sobolev embeddings, to draw conclusions about a.s. continuous differentiability of the sample paths.