Characterization of Besov spaces with dominating mixed smoothness by differences

Besov spaces with dominating mixed smoothness, on the product of the real line and the torus as well as bounded domains, are studied. A characterization of these function spaces in terms of differences is provided. Applications to random fields, like Gaussian fields and the stochastic heat equation, are discussed, based on a Kolmogorov criterion for Besov regularity with dominating mixed smoothness.