Shift-generated $\alpha$-homogeneous classes of jointly measurable random fields

We consider a class of shift-generated alpha-homogeneous random fields (RFs) C[Z] defined through a functional identity involving a fixed positive alpha and a given jointly measurable R^d-valued RF Z(t),t in R^l. The significance of such classes lies in the fact that their elements generate max-stable and stationary RFs. We extend the original functional identity to a broad class of functionals, including the integral operator S(.) and prove that C[Z] contains at least one L^alpha-continuous element. Finally, we investigate properties of local RFs and their connections with spectral tail and tail RFs.