Operator scaling stable random fields

A scalar valued random field {X(x)}x∈Rd{X(x)}x∈Rd is called operator-scaling   if for some d×dd×d matrix EE with positive real parts of the eigenvalues and some H>0H>0 we have{X(cEx)}x∈Rd=f.d.{cHX(x)}x∈Rdfor all c>0, where =f.d. denotes equality of all finite-dimensional marginal distributions. We present a moving average and a harmonizable representation of stable operator scaling random fields by utilizing so called EE-homogeneous functions φφ, satisfying φ(cEx)=cφ(x)φ(cEx)=cφ(x). These fields also have stationary increments and are stochastically continuous. In the Gaussian case, critical Hölder-exponents and the Hausdorff-dimension of the sample paths are also obtained.